^ 


THE 


AMERICAN  HOUSE  CARPENTER. 

A  TREATISE 

ON  THE 

ART  OF  BUILDING. 


COMPRISING 


STYLES  OF  ARCHITECTURE,  STRENGTH  OF  MATERIALS, 


B  D 


THE  THEORY  AND  PRACTICE  OF  THE  CONSTRUCTION  OF   FLOORS,  FRAMED 

GIRDERS,    ROOF  TRUSSES,   ROLLED-IRON  BEAMS,   TUBULAR-IRON 

GIRDKRS,   CAST-IRON   GIRDERS,    STAIRS,    DOORS, 

WINDOWS,  MOULDINGS,  AND  CORNICES; 


A   COMPEND   OF   MATHEMATICS. 


A    MANUAL    FOR    THE    PRACTICAL    USE    OF 

ARCHITECTS,  CARPENTERS,  STAIR-BUILDERS, 

AND    OTHERS. 

EIGHTH     EDITION, 
REWRITTEN    AND    ENLARGED. 

BY 

R.  G.  HATFIELD,  ARCHITECT, 

l\ 

LATE    FELLOW    OF    THE    AMERICAN    INSTITUTE    OF    ARCHITECTS,     MEMBER    OF    THE    AMERICAN 
SOCIETY   OF    CIVIL    ENGINEERS,     ETC. 

AUTHOR    OF    "TRANSVERSE    STRAINS." 

EDITED  BY  O.   P.    HATFIELD,   F.A.I.A.,  ARCHITECT. 
NINTH  EDITION. 

NEW   YORK: 
JOHN    WILEY  &   SONS,    15    ASTOR   PLACE 

1883. 


^/.o 


\ 


COPYRIGHT    1880, 
BY  THE  ESTATE  OF  R.  G.  HAITI  ELD. 


PREFACE. 


SINCE  the  publication  of  the  first  edition  of  this  work,  six  subsequent 
editions  have  been  issued  ;  but,  although  from  time  to  time  many  additions 
to  its  pages  and  revisions  of  its  subject-matter  have  been  made,  still  its  sev- 
eral issues  have  always  been  printed  substantially  from  the  original  stereotype 
plates.  In  this  edition,  however,  the  book  has  been  extensively  remodelled 
and  expanded,  the  greater  portion  of  it  rewritten,  and  the  whole  put  in  a  new 
dress  by  being  newly  set  up  in  type  uniform  in  style  with  that  of  the  late 
author's  recent  work,  Transverse  Strains.  To  this  revision — a  labor  of  love  to 
him — he  devoted  all  the  time  he  could  spare  from  his  other  pressing  engage- 
ments for  a  year  or  more,  and  by  close  and  arduous  application  brought  the 
book  to  a  successful  termination,  notwithstanding  the  engrossing  nature  of 
his  customary  business  avocations.  Although  essentially  an  elementary  work, 
and  intended  originally  for  a  class  of  minds  not  generally  favored  with  oppor- 
tunities for  securing  a  very  extended  form  of  education,  either  in  the  store  of 

*      <r  *  N  x*  N: 

information  acquired  or  in  the  discipline  of  mind  which  culture  confers,  still 
it  has  been  his  aim  to  embody  in  its  pages  so  complete  and  exhaustive  a  treat- 
ment of  the  various  subjects  discussed,  and  so  practical  and  useful  a  collection 
of  data  and  the  rules  governing  their  application,  as  to  make  it  also  not  un- 
worthy the  attention  of  those  who  have  been  more  highly  favored  in  that 
respect. 

In  all  the  various  trades  connected  with  building  it  is  the  intelligent 
workman  that  commands  the  greatest  respect,  and  who  receives  in  all  cases 
the  highest  remuneration.  As  apprentice,  journeyman,  and  master-builder, 
his  course  is  upward  and  onward,  and  success  crowns  his  efforts  in  all  that  he 
undertakes.  There  is  a  kind  of  freemasonry  in  the  very  air  that  surrounds 
the  skilful,  intelligent  man,  that  gives  him  a  pass  at  once  into  the  appreciation 
and  recognition  of  all  those  whose  regard  is  valuable.  We  admire  and  respect 
the  plodding  toil  of  the  honest,  patient  laborer,  whose  humble  task  may  tax 
his  muscles  though  not  his  mind,  but  we  yield  a  deeper  homage  to  the  skilful 
hand  and  tutored  eye  that  accomplish  wonders  in  art  and  science  through  per- 
severance in  aspiring  studies.  It  was  to  excite  in  the  minds  of  workmen  like 
these  an  ambition  to  excel  in  their  calling,  and  to  point  out  to  them  the  surest 
path  to  -that  consummation,  that  the  preparation  of  this  volume  was  under- 
taken ;  that  all  its  tendencies  are  in  that  direction,  and  that  it  cannot  well  fail 


11  PREFACE. 

of  its  purpose  when  judiciously  used,  must  be  the  conviction  of  all  who  will 
take  the  trouble  to  examine  its  pages. 

In  the  first  part  of  the  book  matters  more  particularly  relating  to  building 
are  treated  of.  The  first  section  is  in  the  nature  of  an  introduction,  serving 
by  its  historical  references  to  excite  an  interest  in  the  general  subject,  while 
in  the  second  are  presented  the  methods  of  erecting  edifices  in  accordance 
with  the  acknowledged  principles  of  sound  construction.  In  the  remaining 
sections  of  Part  I.  the  several  well-defined  branches  of  house-building,  as 
stairs,  doors  and  windows,  etc.,  are  illustrated  and  explained.  In  the  second 
part  the  more  useful  rules  and  simple  problems  of  mathematics  are  reduced 
to  an  easily  acquired  form,  and  adapted  to  the  necessities  of  the  ordinary 
workman.  By  studying  the  latter,  the  young  mechanic  may  not  only  improve 
and  strengthen  his  mind,  but  grow  more  self-reliant  daily,  demonstrating  in 
his  own  experience  that  scientific  knowledge  gives  power.  By  carefully 
studying  this  part  of  the  book  he  will  see  how  easy  it  is  to  acquire  the  knowl- 
edge of  solving  problems  by  signs  and  symbols,  commonly  called  Algebra 
(although  looked  upon  by  the  uninitiated  as  almost  incomprehensible),  and 
thus  find  it  easy  to  understand  all  the  illustrations  of  the  various  subjects 
wherein  those  condensed  forms  of  expression  are  used.  Useful  problems  in 
geometry,  described  in  simple^ language,  and  hints  upon  the  subject  of  draw- 
ing and  shading,  are  also  to  be  found  in  Part  II.  A  glossary  of  architectural 
terms  and  many  useful  tables  are  provided  in  the  Appendix,  and  finally,  an 
Index  is  added  to  aid  in  referring  to  special  subjects.  The  full-plate  illustra- 
tions are  inserted  to  make  it  attractive  to  the  general  reader,  and  at  the  same 
time  to  serve  as  explanatory  of  the  historical  portion  of  the  volume. 

It  will  not  be  denied  that  the  class  of  information  herein  furnished  is  one 
of  the  most  instructive  and  useful  that  can.be  presented  to  the  practical  mind 
of  a  workingman,  or  to  any  mind  engaged  in  mechanical  pursuits.  The  im- 
press stamped  upon  it  by  the  author's  peculiar  line  of  study  is  not  to  be 
effaced,  but  this  has  given  it  characteristics  of  originality  and  strength  not 
to  be  found  in  a  mere  compilation. 

THE  EDITOR. 

NEW  YORK,  31  Pine  Street, 
January  6,  1880. 


CONTENTS. 


For  a  Table  of  Contents  more  in  detail,  see  p.  613,  and  for  Index,  see  p.  657. 


PART  I. 

PAGE. 

SECTION  I. — Architecture 5 

II. — Construction 57 

III. — Stairs 240 

"       IV. — Doors  and  Windows 315 

"        V. — Mouldings  and  Cornices 323 


PART  II. 

SECTION  VI.— Geometry 347 

VII. — Ratio  or  Proportion 366 

"      VIII. — Fractions 378 

IX. — Algebra 392 

X.— Polygons 439 

XL— The  Circle 468 

'•       XII.— The  Ellipse 481 

"     XIII.— The  Parabola 492 

"      XIV.— Trigonometry 5IQ 

XV.— Drawing 53^ 

"     XVI. — Practical  Geometry 544 

"    XVII.— Shadows..... S9& 

Table  of  Contents  in  detail . . 6l3 


iv  .CONTENTS. 

APPENDIX. 

Glossary 627 

Table  of  Squares,  Cubes  and  Roots 638 

Rules  for  the  Reduction  of  Decimals 647 

Table  of  Circles 649 

Table  showing  the  Capacity'of  Wells,  Cisterns,  etc 653 

Table  of  the  Weights  of  Materials 654 

Index 657 


((UNIVERSITY 


UNIVERSITY 


PART    I. 


SECTION    L— ARCHITECTURE. 


ART.  I. — Building  Defined. — Building"  and  Architecture 
are  technical  terms  by  some  thought  to  be  synonymous  ; 
but  there  is  a  distinction.  Architecture  has  been  defined  to 
be— "the  art  of  building;"  but  more  correctly  it  is — "the 
art  of  designing  and  constructing  buildings,  in  accordance 
with  such  principles  as  constitute  stability,  utility,  and 
beauty."  The  literal  signification  of  the  Greek  word  arclii- 
tecton,  from  which  the  word  architect  is  derived,  is  chief- 
carpenter  ;  and  the  architect  who  designs  and  builds  well 
may  truly  be  considered  the  chief  builder.  Of  the  three 
classes  into  which  architecture  has  £>een  divided — viz.,  Civil, 
Military,  and  Naval — the  first  is  that  which  refers  to  the 
construction  of  edifices  known  as  dwellings,  churches,  and 
other  public  buildings,  bridges,  etc.,  for  the  accommodation 
of  civilized  man— and  is  the  subject  of  the  remarks  which 
follow.  . 

2.— Antique  Buildings;  Tower  of  Babel.— Building  is 
one  of  the  most  ancient  of  the  arts :  the  Scriptures  inform 
us  of  its  existence  at  a  very  early  period.  Cain,  the  son  of 
Adam,  "  builded  a  city,  and  called  the  name  of  the  city 
after  the  name  of  his  son,  Enoch  ;"  but  of  the  peculiar  style 
or  manner  of  building  we  are  not  informed.  It  is  presumed 
that  it  was  not  remarkable  for  beauty,  but  that  utility  and 
perhaps  stability  were  its  characteristics.  Soon  after  the 
deluge  — that  memorable  event,  which  removed  from  ex- 
istence all  traces  of  the  works  of  man— the  Tower  of  Babel 


6  ARCHITECTURE. 

was  commenced.  This  was  a  work  of  such  magnitude  that 
the  gathering  of  the  materials,  according  to  some  writers, 
occupied  three  years;  the  period  from  its  commencement 
until  the  work  was  abandoned  was  twenty-two  years  ;  and 
the  bricks  were  like  blocks  of  stone,  being  twenty  feet  long, 
fifteen  broad,  and  seven  thick.  Learned  men  have  given  it 
as  their  opinion  that  the  tower  in  the  temple  of  Belus  at 
Babylon  was  the  same  as  that  which  in  the  Scriptures  is 
called  the  Tower  of  Babel.  The  tower  of  the  temple  of 
Belus  was  square  at  its  base,  each  side  measuring  one 
furlong,  and  consequently  half  a  mile  in  circumference.  Its 
form  was  that  of  a  pyramid,  and  its  height  was  660  feet.  It 
had  a  winding  passage  on  the  outside  from  the  base  to  the 
summit,  which  was  wide  enough  for  two  carriages. 

3. — Ancient  Cities  and  Monuments. — Historical  accounts 
of  ancient  cities,  such  as  Babylon,  Palmyra,  and  Nineveh  of 
the  Assyrians ;  Sidon,  Tyre,  Aradus,  and  Serepta  of  the 
Phoenicians;  and  Jerusalem,  with  its  splendid  temple,  of 
the  Israelites  —  show  that  architecture  among  them  had 
made  great  advances.  Ancient  monuments  of  the  art  are 
found  also  among  other  nations  ;  the  subterraneous  temples 
of  the  Hindoos  upon  the  islands  Elephanta  and  Salsetta ; 
the  ruins  of  Persepolis  in  Persia ;  pyramids,  obelisks,  tem- 
ples, palaces,  and  sepulchres  in  Egypt — all  prove  that  the 
architects  of  those  early  times  were  possessed  of  skill  and 
judgment  highly  cultivated.  The  principal  characteristics 
of  their  works  are  gigantic  dimensions,  immovable  solidity, 
and,  in  some  instances,  harmonious  splendor.  The  extra- 
ordinary size  of  some  is  illustrated  in  the  pyramids  of  Egypt. 
The  largest  of  these  stands  not  far  from  the  city  of  Cairo : 
its  base,  which  is  square,  covers  about  ir£  acres,  and  its 
height  is  nearly  500  feet.  The  stones  of  which  it  is  built 
are  immense — the  smallest  being  full  thirty  feet  long. 

4. — Architecture  in  Greece. — Among  the  Greeks,  archi- 
tecture was  cultivated  as  a  fine  art.  Dignity  and  grace 
were  added  to  stability  and  magnificence.  In  the  Doric 
order,  their  first  style  of  building,  this  is  fully  exemplified. 
Phidias,  Ictinus,  and  Calicrates  are  spoken  of  as  masters  in 


GRECIAN   AND    ROMAN   BUILDINGS.  7 

the  art  at  this  period :  the  encouragement  and  support  of 
Pericles  stimulated  them  to  a  noble  emulation.  The  beauti- 
ful temple  of  Minerva,  called  the  Parthenon,  erected  upon 
the  acropolis  of  Athens,  the  Propyleum,  the  Odeum,  and 
others,  were  lasting  monuments  of  their  success.  The  Ionic 
and  Corinthian  orders  were  added  to  the  Doric,  and  many 
magnificent  edifices  arose.  These  exemplified,  in  their 
chaste  proportions,  the  elegant  refinement  of  Grecian  taste. 
Improvement  in  Grecian  architecture  continued  to  advance 
until  perfection  seems  to  have  been  attained.  The  speci- 
mens which  have  been  partially  preserved  exhibit  a  com- 
bination of  elegant  proportion,  dignified  simplicity,  and 
majestic  grandeur.  Architecture  among  the  Greeks  was  at 
the  height  of  its  glory  at  the  period  immediately  preceding 
the  Peloponnesian  war ;  after  which  the  art  declined.  An 
excess  of  enrichment  succeeded  its  former  simple  grandeur  ; 
yet  a  strict  regularity  was  maintained  amid  the  profusion  of 
ornament.  After  the  death  of  Alexander,  323  B.C.,  a  love 
of  gaudy  splendor  increased :  the  consequent  decline  of  the 
art  was  visible,  and  the  Greeks  afterwards  paid  but  little 
attention  to  the  science. 

5. — Architecture  in  Rome. — While  the  Greeks  illustrated 
their  knowledge  of  architecture  in  the  erection  of  their 
temples  and  other  public  buildings,  the  Romans  gave  their 
attention  to  the  science  in  the  construction  of  the  many 
aqueducts  and  sewers  with  which  Rome  abounded  ;  build- 
ing no  such  splendid  edifices  as  adorned  Athens,  Corinth, 
and  Ephesus,  until  about  200  years  B.C.,  when  their  inter- 
course with  the  Greeks  became  more  extended.  Grecian 
architecture  was  introduced  into  Rome  by  Sylla ;  by  whom, 
as  also  by  Marius  and  Caesar,  many  large  edifices  were 
erected  in  various  cities  of  Italy.  But  under  Csesar  Augus- 
tus, at  about  the  beginning  of  the  Christian  era,  the  art 
arose  to  the  greatest  perfection  it  ever  attained  in  Italy. 
Under  his  patronage  Grecian  artists  were  encouraged,  and 
many  emigrated  to  Rome.  It  was  at  about  this  time  that 
Solomon's  temple  at  Jerusalem  was  rebuilt  by  Herod  —  a 
Roman.  This  was  46  years  in  the  erection,  and  was  most 
probably  of  '  the  Grecian  style  of  building— perhaps  of  the 


CATHEDRAL  OF   NOTRE 


THE   GOTHS   AND   VANDALS.  9 

provement ;  but  very  soon  after  his  reign  the  art  began 
rapidly  to  decline,  as  particularly  evidenced  in  the  mean 
and  trifling  character  of  the  ornaments. 

7. — Architecture  Debased. — The  Goths  and  Vandals 
overran  Italy,  Greece,  Asia,  and  Africa,  destroying  most 
of  their  works  of  ancient  architecture.  Cultivating  no  art 
but  that  of  war,  these  savage  hordes  could  not  be  expected 
to  take  any  interest  in  the  beautiful  forms  and  proportions 
of  their  habitations.  From  this  time  architecture  assumed 
an  entirely  different  aspect.  The  celebrated  styles  of  Greece 
were  unappreciated  and  forgotten  ;  and  modern  architec- 
ture made  its  first  appearance  on  the  stage  of  existence. 
The  Goths,  in  their  conquering  invasions,  gradually  ex- 
tended it  over  Italy,  France,  Spain,  Portugal,  and  Ger- 
many, into  England.  From  the  reign  of  Galienus  may  be 
reckoned  the  total  extinction  of  the  arts  among  the  Romans. 
From  this  time  until  the  sixth  or  seventh  century,  architec- 
ture was  almost  entirely  neglected.  The  buildings  which 
were  erected  during  this  suspension  of  the  arts  were  very 
rude.  Being  constructed  of  the  fragments  of  the  edifices 
which  had  been  demolished  by  the  Visigoths  in  their  unre- 
strained fury,  and  the  builders  being  destitute  of  a  proper 
knowledge  of  architecture,  many  sad  blunders  and  exten- 
sive patch-work  might  have  been  seen  in  their  construction 
— entablatures  inverted,  columns  standing  on  their  wrong 
ends,  and  other  ridiculous  arrangements  characterized  their 
clumsy  work.  The  vast  number  of  columns  which  the  ruins 
around  them  afforded  they  used  as  piers  in  the  construction 
of  arcades — which  by  some  is  thought,  after  having  passed 
through  various  changes,  to  have  been  the  origin  of  the 
plan  of  the  Gothic  cathedral.  Buildings  generally,  which 
are  not  of  the  classical  styles,  and  which  were  erected  after 
the  fall  of  the  Roman  empire,  have  by  some  been  indiscrim- 
inately included  under  the  term  Gothic.  But  the  changes 
which  architecture  underwent  during  the  Mediaeval  age 
show  that  there  were  then  several  distinct  modes  of  building. 

8. — The  O§trogoths. — Theodoric,  a  friend  of  the  arts, 
who  reigned  in  Italy  from  A.D.  493  to  525,  endeavored  to 


10  ARCHITECTURE. 

restore  and  preserve  some  of  the  ancient  buildings;  and 
erected  others,  the  ruins  of  which  are  still  seen  at  Verona 
and  Ravenna.  Simplicity  and  strength  are  the  character- 
istics of  the  structures  erected  by  him  ;  they  are,  however, 
devoid  of  grandeur  and  elegance,  or  fine  proportions. 
These  are  properly  of  the  GOTHIC  style  ;  by  some  called 
the  old  Gothic,  to  distinguish  it  from  the  pointed  Gothic. 

9. — The  Lombard*,  who  ruled  in  Italy  from  A.D.  568, 
had  no  taste  for  architecture  nor  respect  for  antiquities. 
Accordingly,  they  pulled  down  the  splendid  monuments  of 
classic  architecture  which  they  found  standing,  and  erected 
in  their  stead  huge  buildings  of  stone  which  were  greatly 
destitute  of  proportion,  elegance,  or  utility — their  charac- 
teristics being  scarcely  anything  more  than  stability  and 
immensity  combined  with  ornaments  of  a  puerile  character. 
Their  churches  were  decorated  with  rows  of  small  columns 
along  the  cornice  of  the  pediment,  small  doors  and  win- 
dows with  circular  heads,  roofs  supported  by  arches  having 
arched  buttresses  to  resist  their  thrust,  and  a  lavish  display 
of  incongruous  ornaments.  This  kind  of  architecture  is 
called  the  LOMBARD  style,  and  was  employed  in  the  seventh 
century  in  Pavia,  the  chief  city  of  the  Lombards  ;  at  which 
city,  as  also  at  many  other  places,  a  great  many  edifices 
were  erected  in  accordance  with  its  peculiar  forms. 

10. — Tlic  Byzantine  Architects,  of  Byzantium,  Constan- 
tinople, erected  many  spacious  edifices;  among  which  are 
included  the  cathedrals  of  Bamberg,  Worms,  and  Mentz, 
and  the  most  ancient  part  of  the  minster  at  Strassburg  ;  in 
all  of  these  they  combined  the  classic  styles  with  the  crude 
Lombardian.  This  style  is  called  the  LOMBARD-BYZANTINE. 
To  the  last  style  there  were  afterwards  added  cupolas  sim- 
ilar to  those  used  in  the  East,  together  with  numerous  slen- 
der pillars  with  elaborate  capitals,  and  the  many  minarets 
which  are  the  characteristics  of  the  proper  Byzantine,  or 
Oriental  style. 

H. — The  Moor*. — When  the  Arabs  and  Moors  destroyed 
the  kingdom  of  the  Goths,  the  arts  and  sciences  were  mostly 


E  LI 

DIVERSITY) 

>,  ^  K          / 


MOSQUE   AT   CAIRO. 


THE   MEDLEVAL   STYLES.  II 

in  possession  of  the  Musselmen-conquerors  ;  at  which  time 
there  were  three  kinds  of  architecture  practised  ;  viz. :  the 
Arabian,  the  Moorish,  and  the  Lombardian.  The  ARABIAN 
style  was  formed  from  Greek  models,  having  circular  arches 
added,  and  towers  which  terminated  with  globes  and  mina- 
rets. The  MOORISH  is  very  similar  to  the  Arabian,  being 
distinguished  from  it  by  arches  in  the  form  of  a  horseshoe. 
It  originated  in  Spain  in  the  erection  of  buildings  with  the 
ruins  of  Roman  architecture,  and  is  seen  in  all  its  splendor 
in  the  ancient  palace  of  the  Mohammedan  monarchs  at 
Grenada,  called  the  Alhambra,  or  red-house.  The  style  which 
was  originated  by  the  Visigoths  in  Spain  by  a  combination 
of  the  Arabian  and  Moorish  styles,  was  introduced  by  Charle- 
magne into  Germany.  On  account  of  the  changes  and  im- 
provements it  there  underwent,  it  Was,  at  about  the  I3th  or 
I4th  century,  termed  the  German  or  romantic  style.  It  is  ex- 
hibited in  great  perfection  in  the  towers  of  the  minster  of 
Strassburg,  the  cathedral  of  Cologne  and  other  edifices. 
The  most  remarkable  features  of  this  lofty  and  aspiring  style 
are  the  lancet  or  pointed  arch,  clustered  pillars,  lofty  towers, 
and  flying  buttresses.  It  was  principally  employed  in  eccle- 
siastical architecture,  and  in  this  capacity  introduced  into 
France,  Italy,  Spain,  and  England. 

12. — Ttie  Architecture  of  England:  is  divided  into  the 
Norman,  the  Early-English,  the  Decorated,  and  the  Perpendic- 
ular styles.  The  Norman  is  principally  distinguished  by 
the  character  of  its  ornaments — the  chevron,  or  zigzag,  being 
the  most  common.  Buildings  in  this  style  were  erected  in 
the  1 2th  century.  The  Early-English  is  celebrated  for  the 
beauty  of  its  edifices,  the  chaste  simplicity  and  purity  of 
design  which  they  display,  and  the  peculiarly  graceful  char- 
acter of  its  foliage.  This  style  is  of  the  isth  century.  The 
Decorated  style,  as  its  name  implies,  is  characterized  by  a 
great  profusion  of  enrichment,  which  consists  principally  of 
the  crocket,  or  feathered-ornament,  and  ball-flower.  It  was 
mostly  in  use  in  the  Hth  century.  The  Perpendicular  style, 
which  dates  from  the  I5th  century,  is  distinguished  by  its 
high  towers,  and  parapets  surmounted  with  spires  similar  in 
number  and  grouping  to  oriental  minarets. 


12  ARCHITECTURE. 

13.— Architecture  Progresiiive.— The  styles  erroneously 
termed  Gothic  were  distinguished  by  peculiar  characteris- 
tics as  well  as  by  different  names.  The  first  symptoms  of  a 
desire  to  return  to  a  pure  style  in  architecture,  after  the 
ruin  caused  by  the  Goths,  was  manifested  in  the  character 
of  the  art  as  displayed  in  the  church  of  St.  Sophia  at  Con- 
stantinople, which  was  erected  by  Justinian  in  the  6th 
century.  The  church  of  St.  Mark  at  Venice,  which  arose 
in  the  loth  or  nth  century,  is  a  most  remarkable  building; 
a  compound  of  many  of  the  forms  of  ancient  architecture. 
The  cathedral  at  Pisa,  a  wonderful  structure  for  the  age, 
was  erected  by  a  Grecian  architect  in  1016.  The  marble 
with  which  the  walls  of  this  building  were  faced,  and  of 
which  the  four  rows  of  columns  that  support  the  roof  are 
composed,  is  said  to  be  of  an  excellent  character.  The 
Campanile,  or  leaning-tower  as  it  is  usually  called,  was 
erected  near  the  cathedral  in  the  I2th  century.  Its  inclina- 
tion is  generally  supposed  to  have  arisen  from  a  poor  foun- 
dation ;  although  by  some  it  is  said  to  have  been  thus  con- 
structed originally,  in  order  to  inspire  in  the  minds  of  the 
beholder  sensations  of  sublimity  and  awe.  In  the  I3th  cen- 
tury, the  science  in  Italy  was  slowly  progressing  ;  many  fine 
churches  were  erected,  the  style  of  which  displayed  a  de- 
cided advance  in  the  progress  towards  pure  classical  archi- 
tecture. In  other  parts  of  Europe,  the  Gothic,  or  pointed 
style  was  prevalent.  The  cathedral  at  Strassburg,  designed 
by  Irwin  Steinbeck,  was  erected  in  the  I3th  and  I4th  cen- 
turies. In  France  and  England  during  the  I4th  century, 
many  very  superior  edifices  were  erected  in  this  style. 

14-. — Architecture  in  Italy. — In  the  I4th  and  1 5th  cen- 
turies, architecture  in  Italy  was  greatly  revived.  The  mas- 
ters began  to  study  the  remains  of  ancient  Roman  edifices  ; 
and  many  splendid  buildings  were  erected,  which  displayed 
a  purer  taste  in  the  science.  Among  others,  St.  Peter's  of 
Rome,  which  was  built  about  this  time,  is  a  lasting  monu- 
ment of  the  architectural  skill  of  the  age.  Giocondo,  Mi- 
chael Angelo,  Palladio,  Vignola,  and  other  celebrated  archi- 
tects, each  in  their  turn,  did  much  to  restore  the  art  to  its 


INTERIOR    OF    ST 


SOPHIA,    CONSTANTINOPLE. 


-gjpl  Lift 

Y^ 

UNIVERSITY] 


ORIGIN   OF   STYLES.  13 

former  excellence.  In  the  edifices  which  were  erected  under 
their  direction,  however,  it  is  plainly  to  be  seen  that  they 
studied  not  from  the  pure  models  of  Greece,  but  from  the 
remains  of  the  deteriorated  architecture  of  Rome.  The 
high  pedestal,  the  coupled  columns,  the  rounded  pediment, 
the  many  curved-and-twisted  enrichments,  and  the  convex 
frieze,  were  unknown  to  pure  Grecian  architecture.  Yet 
their  efforts  were  serviceable  in  correcting,  to  a  good  de- 
gree, the  very  impure  taste  that  had  prevailed  since  the  over- 
throw of  the  Roman  empire. 

15. — The  Renaissance. — The  Italian  masters  and  numer- 
ous artists  who  had  visited  Italy  for  the  purpose,  spread  the 
Roman  style  over  various  countries  of  Europe;  which  was 
gradually  received  into  favor  in  place  of  the  pointed  Gothic. 
This  fell  into  disuse ;  although  it  has  of  late  years  been 
again  cultivated.  It  requires  a  building  of  great  magnitude 
and  complexity  for  a  perfect  display  of  its  beauties.  In 
America,  the  pure  Grecian  style  was  at  first  more  or  less 
studied ;  and  perhaps  the  simplicity  of  its  principles  would 
be  better  adapted  to  a  republican  country  than  the  more 
intricate  mediaeval  styles  ;  yet  these,  during  the  last  quarter 
of  a  century,  have  been  extensively  studied,  and  now  wholly 
supersede  the  Grecian  styles. 

16. — Style§  of  Arehiteeture. — It  is  generally  acknowl- 
edged that  the  various  styles  in  architecture  were  the  results 
of  necessity,  and  originated  in  accordance  with  the  different 
pursuits  of  the  early  inhabitants  of  the  earth  ;  and  were 
brought  by  their  descendants  to  their  present  state  of  per- 
fection, through  the  propensity  for  imitation  and  desire  of 
emulation  which  are  found  more  or  less  among-  all  nations. 
Those  that  followed  agricultural  pursuits,  from  being  em- 
ployed constantly  upon  the  same  piece  of  land,  needed  a 
permanent  residence,  and  the  wooden  hut  was  the  offspring 
of  their  wants  ;  while  the  shepherd,  who  followed  his  flocks 
and  was  compelled  to  traverse  large  tracts  of  country  for 
pasture,  found  the  tent  to  be  the  most  portable  habitation ; 
again,  the  man  devoted  to  hunting  and  fishing — an  idle  and 
vagabond  way  of  living — is  naturally  supposed  to  have  been 


14  ARCHITECTURE. 

content  with  the  cavern  as  a  place  of  shelter.  .The  latter  is 
said  to  have  been  the  origin  of'the  Egyptian  style;  while 
the  curved  roof  of  Chinese  structures  gives  a  strong  indica- 
tion of  their  having  had  the  tent  for  their  model  ;  and  the 
simplicity  of  the  original  style  of  the  Greeks  (the  Doric) 
shows  quite  conclusively,  as  is  generally  conceded,  that  its 
original  was  of  wood.  The  pointed,  or  ecclesiastical  style, 
is  said  to  have  originated  in  an  attempt  to  imitate  the  bower, 
or  grove  of  trees,  in  which  the  ancients  performed  their  idol- 
worship.  But  it  is  more  probably  the  result  of  repeated 
scientific  attempts  to  secure  real  strength  with  apparent 
lightness ;  thus  giving  a  graceful,  aspiring  effect. 

17.— Order§:  or  styles,  in  architecture  are  numerous; 
and  a  knowledge  of  the  peculiarities  of  each  is  important  to 
the  student  in  the  art.  An  ORDER,  in  architecture,  is  com- 
posed of  three  principal  parts,  viz. :  the  Stylobate,  the  Col- 
umn, and  the  Entablature.  This  appertains  chiefly  to  the 
classic  styles. 

18. — The  Stylobate:  is  the  substructure,  or  basement, 
upon  which  the  columns  of  an  order  are  arranged.  In 
Roman  architecture — especially  in  the  interior  of  an  edi- 
fice— it  frequently  occurs  that  each  column  has  a  separate 
substructure  ;  this  is  called  a  pedestal.  If  possible,  the  ped- 
estal should  be  avoided  in  all  cases ;  because  it  gives  to  the 
column  the  appearance  of  having  been  originally  designed 
for  a  small  building,  and  afterwards  pieced  out  to  make  it 
long  enough  for  a  larger  one. 

19. — The  Column :  is  composed  of  the  base,  shaft,  and 
capital. 

20. — Tlie  Entablature:  above  and  supported  by  the 
columns,  is  horizontal ;  and  is  composed  of  the  architrave, 
frieze,  and  cornice.  These  principal  parts  are  again  divided 
into  various  members  and  mouldings. 

21. — The  Base:  of  a  column  is  so  called  from  basis,  a 
foundation  or  footing. 


^t°        ^"4j>2\ 

7A    > 

UNIVERSITY) 


INTERIOR    OF    ST.    STEPHENS,    PARIS. 


PARTS   OF  AN   ORDER.  15 

22.— The  Shaft:  the  upright  part  of  a  column  standing 
upon  the  base  and  crowned  with  the  capital,  is  from  shafto, 
to  dig — in  the  manner  of  a  well,  whose  inside  is  not  unlike 
the  form  of  a  column. 

23. — The  Capital :  from  kephale  or  caput,  the  head,  is  the 
uppermost  and  crowning  part  of  the  column. 

24. — The  Architrave  :  from  archi,  chief  or  principal, 
and  trabs,  a  beam,  is  that  part  of  the  entablature  which  lies 
in  immediate  connection  with  the  column. 

25. — The  Frieze:  from  fibron,  a  fringe  or  border,  is  that 
part  of  the  entablature  which  is  immediately  above  the 
architrave  and  beneath  the  cornice.  It  was  called  by  some 
of  the  ancients  zophorus,  because  it  was  usually  enriched 
with  sculptured  animals. 

26. — The  Corniee:  from  corona,  a  crown,  is  the  upper 
and  projecting  part  of  the  entablature — being  also  the  upper- 
most and  crowning  part  of  the  whole  order. 

27. — The  Pediment:  above  the  entablature,  is  the  tri- 
angular portion  which  is  formed  by  the  inclined  edges  of 
the  roof  at  the  end  of  the  building.  In  Gothic  architecture, 
the  pediment  is  called  a  gable. 

28. — The  Tympanum:  is  the  perpendicular  triangular 
surface  which  is  enclosed  by  the  cornice  of  the  pediment. 

29. — The  Attic :  is  a  small  order,  consisting  of  pilasters 
and  entablature,  raised  above  a  larger  order,  instead  of  a 
pediment.  An  attic  story  is  the  upper  story,  its  windows 
being  usually  square. 

30. — Proportions  in  an  Order. — An  order  has  its  several 
members  proportioned  to  one  another  by  a  scale  of  60  equal 
parts,  which  are  called  minutes.  If  the  height  of  buildings 
were  always  the  same,  the  scale  of  equal  parts  would  be  a 
fixed  quantity— an  exact  number  of  feet  and  inches.  But  as 
buildings  are  erected  of  different  heights,  the  column  and 


1 6  ARCHITECTURE. 

its  accompaniments  are  required  to  be  of  different  dimen- 
sions. To  ascertain  the  scale  of  equal  parts,  it  is  necessary 
to  know  the  height  to  which  the  whole  order  is  to  be 
erected.  This  must  be  divided  by  the  number  of  diameters 
which  is  directed  for  the  order  under  consideration.  Then 
the  quotient  obtained  by  such  division  is  the  length  of  the 
scale  of  equal  parts — and  is,  also,  the  diameter  of  the  column 
next  above  the  base.  For  instance,  in  the  Grecian  Doric 
order  the  whole  height,  including  column  and  entablature, 
is  8  diameters.  Suppose  now  it  were  desirable  to  construct 
an  example  of  this  order,  forty  feet  high.  Then  40  feet 
divided  by  8  gives  5  feet  for  the  length  of  the  scale  ;  and 
this  being  divided  by  60,  the  scale  is  completed.  The  up- 
right columns  of  figures,  marked  H  and  P,  by  the  side  of 
the  drawings  illustrating  the  orders,  designate  the  height 
and  the  projection  of  the  members.  The  projection  of  each 
member  is  reckoned  from  a  line  passing  through  the  axis  of 
the  column,  and  extending  above  it  to  the  top  of  the  entab- 
lature. The  figures  represent  minutes,  or  6oths,  of  the 
major  diameter  of  the  shaft  of  the  column. 

31. — Grecian  Styles.— The  original  method  of  building 
among  the  Greeks  was  in  what  is  called  the  Doric  order : 
to  this  were  afterwards  added  the  Ionic  and  the  Corinthian. 
These  three  were  the  only  styles  known  among  them.  Each 
is  distinguished  from  the  other  two  by  not  only  a  peculiar- 
ity of  some  one  or  more  of  its  principal  parts,  but  also  by  a 
particular  destination.  The  character  of  the  Doric  is  robust, 
manly,  and  Herculean-like  ;  that  of  the  Ionic  is  more  deli- 
cate, feminine,  matronly;  while  that  of  the  Corinthian  is 
extremely  delicate,  youthful,  and  virgin-like.  However 
they  may  differ  in  their  general  character,  they  are  alike 
famous  for  grace  and  dignity,  elegance  and  grandeur,  to  a 
high  degree  of  perfection. 

32 — The  Doric  Order:  (Fig.  2,)  is  so  ancient  that  its 
origin  is  unknown— although  some  have  pretended  to  have 
discovered  it.  But  the  most  general  opinion  is,  that  it  is 
an  improvement  upon  the  original  wooden  buildings  of  the 


FANCIFUL   ORIGIN   OF   THE   DORIC.  I/ 

Grecians.     These  no  doubt  were  very  rude,  and  perhaps 
not  unlike  the  following  figure. 


FIG   i. — SUPPOSED  ORIGIN  OF  DORIC  TEMPLE. 

The  trunks  of  trees,  set  perpendicularly  to  support  the 
roof,  may  be  taken  for  columns  ;  the  tree  laid  upon  the 
tops  of  the  perpendicular  ones,  the  architrave  ;  the  ends 
of  the  cross-beams  which  rest  upon  the  architrave,  the 
triglyphs  ;  the  tree  laid  on  the  cross-beams  as  a  support  for 
the  ends  of  the  rafters,  the  bed-moulding  of  the  cornice  ;  the 
ends  of  the  rafters  which  project  beyond  the  bed-moulding, 
the  mutules ;  and  perhaps  the  projection  of  the  roof  in 
front,  to  screen  the  entrance  from  the  weather,  gave  origin 
to  the  portico. 

The  peculiarities  of  the  Doric  order  are  the  triglyphs — 
those  parts  of  the  frieze  which  have  perpendicular  channels 
cut  in  their  surface  ;  the  absence  of  a  base  to  the  column — 
as  also  of  fillets  between  the  flutings  of  the  column  ;  and  the 
plainness  of  the  capital.  The  triglyphs  should  be  so  dis- 
posed that  the  width  of  the  metopes — the  space  between 
the  triglyphs — shall  be  equal  to  their  height. 


33. — The  Intercolumniation  :  or  space  between  the  col- 
umns, is  regulated  by  placing  the  centres  of  the  columns 
under  the  centres  of  the  triglyphs — except  at  the  angle  of 
the  building  ;  where,  as  may  be  seen  in  Fig.  2,  one  edge  of 


18 


ARCHITECTURE. 


FIG.  2. — GRECIAN  DORIC. 


PECULIARITIES   OF  THE   DORIC.  19 

the  triglyph  must  be  over  the  centre  of  the  column.* 
Where  the  columns  are  so  disposed  that  one  of  them  stands 
beneath  every  other  triglyph,  the  arrangement  is  called 
mono-triglyph  and  is  most  common.  When  a  column  is 
placed  beneath  every  third  triglyph,  the  arrangement  is 
called  diastyle  ;  and  when  beneath  every  fourth,  arceostyle. 
This  last  style  is  the  worst,  and  is  seldom  adopted. 

34-.—  The  Doric  Order:  is  suitable  for  buildings  that 
are  destined  for  national  purposes,  for  banking-houses,  etc. 
Its  appearance,  though  massive  and  grand,  is  nevertheless 
rich  and  graceful.  The  Patent  Office  at  Washington,  and 
the  Treasury  at  New  York,  are  good  specimens  of  this 
order. 

35.  —  The  Ionic  Order.  (Fig.  3.)  —  The  Doric  was  for 
some  time  the  only  order  in  use  among  the  Greeks.  They 
gave  their  attention  to  the  cultivation  of  it,  until  perfection 
seems  to  have  been  attained.  Their  temples  were  the  prin- 

*  GRECIAN  DORIC  ORDER.  When  the  width  to  be  occupied  by  the  whole  front 
is  limited,  to  determine  the  diameter  of  the  column. 

The  relation  between  the  parts  may  be  expressed  thus  : 

_  60  a 

~~  ~d(b  '+  c)  +  (60  —  c) 

Where  a  equals  the  width  in  feet  occupied  by  the  columns,  and  their  inter- 
columniations  taken  collectively,  measured  at  the  base  ;  b  equals  the  width 
of  the  metope,  in  minutes  ;  c  equals  the  width  of  the  triglyphs  in  minutes  ;  d 
equals  the  number  of  metopes,  and  x  equals  the  diameter  in  feet. 

Example.  —  A  front  of  six  columns  —  hexastyle  —  61  feet  wide  ;  the  frieze 
having  one  triglyph  over  each  intercolumniation,  or  mono-triglyph.  In  this 
case,  there  being  five  intercolumniations  and  two  metopes  over  each,  therefore 
there  are  5  x  2  =  10  metopes.  Let  the  metope  equal  42  minutes  and  the 
triglyph  equal  28.  Then  a  =  61  ;  b  =  42  ;  c  =  28  ;  and  d  =  10  ;  and  the  formula 
above  becomes 

60  x  61  60  x  61  3660 

x  —  —.  ---  -—  -  --  -  -  ST  =  -  =  -  —  =  5   feet  =  the    d  lameter 
10(42  +  28)  +  (60  —  28)      10x70  +  32        732 

required. 

Example.  —  An  octastyle  front,  8  columns,  184  feet  wide,  three  metopes 
over  each,  intercolumniation,  21  in  all,  and  the  metope  and  triglyph  42  and 
28,  as  before.  Then 


l84  -  =  H212  =  7.35-rigir  feet  =  the  diameter  required. 


21  (42  +  28)  +  (60  -  28)         1502 


20  ARCHITECTURE. 

cipal  objects  upon  which  their  skill  in  the  art  was  displayed  ; 
and  as  the  Doric  order  seems  to  have  been  well  fitted,  by  its 
massive  proportions,  to  represent  the  character  of  their 
male  deities  rather  than  the  female,  there  seems  to  have 
been  a  necessity  for  another  style  which  should  be  emble- 
matical of  feminine  graces,  and  with  which  they  might 
decorate  such  temples  as  were  dedicated  to  the  goddesses. 
Hence  the  origin  of  the  Ionic  order.  This  was  invented, 
according  to  historians,  by  Hermogenes  of  Alabanda ;  and 
he  being  a  native  of  Caria,  then  in  the  possession  of  the 
lonians,  the  order  was  called  the  Ionic. 

The  distinguishing  features  of  this  order  are*  the  volutes 
or  spirals  of  the  capital ;  and  the  dentils  among  the  bed- 
mouldings  of  the  cornice:  although  in  some  instances 
dentils  are  wanting.  The  volutes  are  said  to  have  been 
designed  as  a  representation  of  curls  of  hair  on  the  head  of 
a  matron,  of  whom  the  whole  column  is  taken  as  a  sem- 
blance. 

The  Ionic  order  is  appropriate  for  churches,  colleges, 
seminaries,  libraries,  all  edifices  dedicated  to  literature  and 
the  arts,  and  all  places  of  peace  and  tranquillity.  The  front 
of  the  Custom-House,  New  York  City,  is  a  good  specimen 
of  this  order. 

36. — The  Intercolumniation :  of  this  and  the  other 
orders — both  Roman  and  Grecian,  with  the  exception  of 
the  Doric — are  distinguished  as  follows.  When  the  interval 
is  one  and  a  half  diameters,  it  is  called  pycnostyle,  or  columns 
thick-set;  when  two  diameters,  systyle ;  when  two  and  a 
quarter  diameters,  eustyle  ;  when  three  diameters,  diastyle  ; 
and  when  more  than  three  diameters,  arceostyle,  or  columns 
thin-set.  In  all  the  orders,  when  there  are  four  columns  in 
one  row,  the  arrangement  is  called  tetrastyle ;  when  there 
are  six  in  a  row,  hcxastyle  ;  and  when  eight,  octastyle. 

37.— To  Describe  the  Ionic  Volute.— Draw  a  perpen- 
dicular from  a  to  s  (Fig.  4),  and  make  a  s  equal  to  20  min. 
or  to  $  of  the  whole  height,  a  c  ;  draw  s  o  at  right  angles  to 
s  a,  and  equal  to  I  £  min. ;  upon  o,  with  2^  min.  for  radius, 


PROPORTIONS    OF    GRECIAN    IONIC. 

v 


FIG.  3.— GRECIAN  IONIC. 


22 


ARCHITECTURE. 


describe  the  eye  of  the  volute ;  about  o,  the  centre  of  the 
eye,  draw  the  square,  r  t  i  2,  with  sides  equal  to  half  the 
diameter  of  the  eye,  viz.  2j  min.,  and  divide  it  into  144  equal 
parts,  as  shown  at  Fig.  5.  The  several  centres  in  rotation  are 
at  the  angles  formed  by  the  heavy  lines,  as  figured,  i,  2,  3, 
4,  5,  6,  etc.  The  position  of  these  angles  is  determined  by 
commencing  at  the  point,  i,  and  making  each  heavy  line  one 
part  less  in  length  than  the  preceding  one.  No.  i  is  the 


FIG.  4. — IONIC  VOLUTE. 


THE   IONIC  VOLUTE.  23 

centre  for  the  arc  a  b  (Fig.  4 ;)  2  is  the  centre  for  the  arc 
be;  and  so  on  to  the  last.  The  inside  spiral  line  is  to  be 
described  from  the  centres,  x,  x,  x,  etc.  (Fig.  5),  being  the 
centre  of  the  first  small  square  towards  the  middle  of  the 
eye  from  the  centre  for  the  outside  arc.  The  breadth  of  the 
fillet  at  aj  is  to  be  made  equal  to  2T3¥  min.  This  is  for  a  spiral 
of  three  revolutions ;  but  one  of  any  number  of  revolutions, 
as  4  or  6,  may  be  drawn,  by  dividing  of  (Fig.  5)  into  a  cor- 
responding number  of  equal  parts.  Then  divide  the  part 
nearest  the  centre,  o,  into  two  parts,  as  at  h  ;  join  o  and  i, 
also  o  and  2  ;  draw  h  3  parallel  to  o  i,  and  h  4  parallel  to  o 


FIG.  5. — EYE  OF  VOLUTE. 

2  ;  then  the  lines  o  i,  o  2,  //  3,  h  4  will  determine  the  length 
of  the  heavy  lines,  and  the  place  of  the  centres.  (See  Art. 
288.) 

38.— The  Corinthian  Order :  (Fig.  /,)  is  in  general  like 
the  Ionic,  though  the  proportions  are  lighter.     The  Corin- 
thian displays  a  more  airy  elegance,  a  richer  appearance  ; 
but  its  distinguishing  feature  is  its  beautiful  capital, 
is  generally  supposed  to  have  had  its  origin  in  the  capitals 


24  ARCHITECTURE. 

of  the  columns  of  Egyptian  temples,  which,  though  not  ap- 
proaching it  in  elegance,  have  yet  a  similarity  of  form  with 
the  Corinthian.  The  oft-repeated  story  of  its  origin  which 

is  told  by  Vitruvius — an  architect 
who  flourished  in  Rome  in  the  days 
of  Augustus  Caesar — though  pretty 
generally  considered  to  be  fabu- 
lous, is  nevertheless  worthy  of  be- 
ing again  recited.  It  is  this :  A 
young  lady  of  Corinth  was  sick,  and 
finally  died.  Her  nurse  gathered 
,  into  a  deep  basket  such  trinkets  and 
keepsakes  as  the  lady  had  been 

fond  of  when  alive,  and  placed  them  upon  her  grave,  cover- 
ing the  basket  with  a  flat  stone  or  tile,  that  its  contents 
might  not  be  disturbed.  The  basket  was  placed  accident- 
ally upon  the  stem  of  an  acanthus  plant,  which,  shooting 
forth,  enclosed  the  basket  with  its  foliage,  some  of  which, 
reaching  the  tile,  turned  gracefully  over  in  the  form  of  a 
volute. 

A  celebrated  sculptor,  Calimachus,  saw  the  basket  thus 
•decorated,  and  from  the  hint  which  it  suggested  conceived 
and  constructed  a  capital  for  a  column.  This  was  called 
Corinthian,  from  the  fact  that  it  was  invented  and  first  made 
use  of  at  Corinth. 

The  Corinthian  being  the  gayest,  the  richest,  the  most 
lovely  of  all  the  orders,  it  is  appropriate  for  edifices  which 
are  dedicated  to  amusement,  banqueting,  and  festivity — for 
all  places  where  delicacy,  gavety,  and  splendor  are  desir- 
able. 

39. — Pcr§ian§  and  Caryatides. — In  addition  to  the  three 
regular  orders  of  architecture,  it  was  customary  among  the 
Greeks  and  other  nations  to  employ  representations  of  the 
human  form,  instead  of  columns,  to  support  entablatures ; 
these  were  called  Persians  and  Caryatides. 

40. — Persian* :  are  statues  of  men,  and  are  so  called  in 
commemoration  of  a  victory  gained  over  the  Persians  by 
Pausanias.  The  Persian  prisoners  were  brought  to  Athens 


PROPORTIONS    OF   GRECIAN    CORINTHIAN. 


•x 


sl 


- 


J 


FIG.  "j. — GRECIAN  CORINTHIAN. 


26  ARCHITECTURE. 

and  condemned  to  abject  slavery  ;  and  in  order  to  represent . 
them  in  the  lowest  state  of  servitude  and  degradation,  the 
statues  were  loaded  with  the  heaviest  entablature,  the  Doric. 

41. — Caryatides:  are  statues  of  women  dressed  in  long 
robes  after  the  Asiatic  manner.  Their  origin  is  as  follows : 
In  a  war  between  the  Greeks  and  the  Caryans,  the  latter 
were  totally  vanquished,  their  male  population  extinguished, 
and  their  females  carried  to  Athens.  To  perpetuate  the 
memory  of  this  event,  statues  of  females,  having  the  form 
and  dress  of  the  Caryans,  were  erected,  and  crowned  with 
the  Ionic  or  Corinthian  entablature.  The  caryatides  were 
generally  formed  of  about  the  human  size,  but  the  persians 
much  larger,  in  order  to  produce  the  greater  awe  and 
astonishment  in  the  beholder.  The  entablatures  were  pro- 
portioned to  a  statue  in  like  manner  as  to  a  column  of  the 
same  height. 

These  semblances  of  slavery  have  been  in  frequent  use 
among  moderns  as  well  as  ancients  ;  and,  as  a  relief  from 
the  stateliness  and  formality  of  the  regular  orders,  are  capa- 
ble of  forming  a  thousand  varieties  ;  yet  in  a  land  of  liberty 
such  marks  of  human  degradation  ought  not  to  be  perpetu- 
ated. 

42. — Roman  Styles. — Strictly  speaking,  Rome  had  no 
architecture  of  her  own  ;  all  she  possessed  was  borrowed 
from  other  nations.  Before  the  Romans  exchanged  inter- 
course with  the  Greeks,  they  possessed  some  edifices  of 
considerable  extent  and  merit,  which  were  erected  by  archi- 
tects from  Etruria ;  but  Rome  was  principally  indebted  to 
Greece  for  what  she  acquired  of  the  art.  Although  there  is 
no  such  thing  as  an  architecture  of  Roman  invention,  yet 
no  nation,  perhaps,  ever  was  so  devoted  to  the  cultivation 
of  the  art  as  the  Roman.  Whether  we  consider  the  number 
and  extent  of  their  structures,  or  the  lavish  richness  and 
splendor  with  which  they  were  adorned,  we  are  compelled 
to  yield  to  them  our  admiration  and  praise.  At  one  time, 
under  the  consuls  and  emperors,  Rome  employed  400  ar- 
chitects. The  public  works  —  such  as  theatres,  circuses, 
baths,  aqueducts,  etc.  —  were,  in  extent  and  grandeur,  be- 


PORTICO  OF   THE   ERECTHEUM,    ATHENS. 


CHANGE   OF   STYLES   BY  THE   ROMANS.  2? 

yond  anything-  attempted  in  modern  times.  Aqueducts 
were  built  to  convey  water  from  a  distance  of  60  miles  or 
more.  In  the  prosecution  of  this  work  rocks  and  mountains 
were  tunnelled,  and  valleys  bridged.  Some  of  the  latter 
descended  200  feet  below  the  level  of  the  water;  and.  in 
passing  them  the  canals  were  supported  by  an  arcade,  or 
succession  of  arches.  Public  baths  are  spoken  of  as  large  as 
cities,  being  fitted  up  with  numerous  conveniences  for  ex- 
ercise and  amusement.  Their  decorations  were  most  splen- 
did ;  indeed,  the  exuberance  of  the  ornaments  alone  was 
offensive  to  good  taste.  So  overloaded  with  enrichments 
were  the  baths  of  Diocletian  that  on  one  occasion  of  public 
festivity  great  quantities  of  sculpture  fell  from  the  ceilings 
and  entablatures,  killing  many  of  the  people. 

43. — Grecian   Order§    modified  by  the   Romans. — The 

orders  of  Greece  were  introduced  into  Rome  in  all  their 
perfection.  But  the  luxurious  Romans,  not  satisfied  with 
the  simple  elegance  of  their  refined  proportions,  sought  to 
improve  upon  them  by  lavish  displays  of  ornament.  They 
transformed  in  many  instances  the  true  elegance  of  the 
Grecian  art  into  a  gaudy  splendor,  better  suited  to  their 
less  refined  taste.  The  Romans  remodelled  each  of  the 
orders :  the  Doric  (Fig.  8)  was  modified  by  increasing  the 
height  of  the  column  to  8  diameters ;  by  changing  the 
echinus  of  the  capital  for  an  ovolo,  or  quarter-round,  and 
adding  an  astragal  and  neck  below  it ;  by  placing  the  centre, 
instead  of  one  edge,  of  the  first  triglyph  over  the  centre  of 
the  column  ;  and  introducing  horizontal  instead  of  inclined 
mutules  in  the  cornice,  and  in  some  instances  dispensing 
with  them  altogether.  The  Ionic  was  modified  by  diminish- 
ing the  size  of  the  volutes,  and,  in  some  specimens,  intro- 
ducing a  new  capital  in  which  the  volutes  were  diagonally 
arranged  (Fig.  9).  This  new  capital  has  been  termed  modern 
Ionic.  The  favorite  order  at  Rome  and  her  colonies  was 
the  Corinthian  (Fig.  10).  But  this  order  the  Roman  artists, 
in  their  search  for  novelty,  subjected  to  many  alterations — 
especially  in  the  foliage  of  its  capital.  Into  the  upper  part 
of  this  they  introduced  the  modified  Ionic  capital ;  thus 


28 


ARCHITECTURE. 


combining  the  two  in  one.     This  change  was  dignified  with 
the  importance  of  an  order,  and  received  the  appellation 


+  « 


[29. 


Vft 


' 


j^'* 


n»  >»<y 


x>L->xL 


FIG.  8.— ROMAN  DORIC. 

of  COMPOSITE,  or  Roman :  the  best  specimen  of  which  is 
found  in  the  Arch  of  Titus  (Fig.  n).     This  style  was  not 


PROPORTIONS  OF  THE   ROMAN   IONIC. 


29 


If4 


.-._ 


aygpg^^ 


]^>34&*^&&1&1>S<1&1&Z> 


wmm: 


FIG,  9. — ROMAN  IONIC. 


30  ARCHITECTURE. 

much    used   among    the    Romans   themselves,   and   is   but 
slightly  appreciated  now. 

44. — Tlie  Tuscan  Order:  is  said  to  have  been  intro- 
duced to  the  Romans  by  the  Etruscan  architects,  and  to 
have  been  the  only  style  used  in  Italy  before  the  introduc- 
tion of  the  Grecian  orders.  However  this  may  be,  its  simi- 
larity to  the  Doric  order  gives  strong  indications  of  its 
having  been  a  rude  imitation  of  that  style  :  this  is  very  prob- 
able, since  history  informs  us  that  the  Etruscans  held  inter- 
course with  the  Greeks  at  a  remote  period.  The  rudeness 
of  this  order  prevented  its  extensive  use  in  Italy.  All  that 
is  known  concerning  it  is  from  Vitruvius,  no  remains  of 
buildings  in  this  style  being  found  among  ancient  ruins. 

For  mills,  factories,  markets,  barns,  stables,  etc.,  where 
utility  and  strength  are  of  more  importance  than  beauty, 
the  improved  modification  of  this  order,  called  the  modern 
Tuscan  (Fig.  12),  will  be  useful;  and  its  simplicity  recom- 
mends it  where  economy  is  desirable. 

45. — Egyptian  Style. — The  architecture  of  the  ancient 
Egyptians — to  which  that  of  the  ancient  Hindoos  bears 
some  resemblance — is  characterized  by  boldness  of  outline, 
solidity,  and  grandeur.  The  amazing  labyrinths  and  exten- 
sive artificial  lakes,  the  splendid  palaces  and  gloomy  ceme- 
teries, the  gigantic  pyramids  and  towering  obelisks,  of  the 
Egyptians  were  works  of  immensity  and  durability  ;  and 
their  extensive  remains  are  enduring  proofs  of  the  enlight- 
ened skill  of  this  once-powerful  but  long  since  extinct  na- 
tion. The  principal  features  of  the  Egyptian  style  of  archi- 
tecture are — uniformity  of  plan,  never  deviating  from  right 
lines  and  angles ;  thick  walls,  having  the  outer  surface 
slightly  deviating  inwardly  from  the  perpendicular ;  the 
whole  building  low ;  roof  flat,  composed  of  stones  reaching 
in  one  piece  from  pier  to  pier,  these  being  supported  by 
enormous  columns,  very  stout  in  proportion  to  their  height ; 
the  shaft  sometimes  polygonal,  having  no  base  but  with  a 
great  variety  of  handsome  capitals,  the  foliage  of  these  being 
of  the  palm,  lotus,  and  other  leaves ;  entablatures  having 
simply  an  architrave,  crowned  with  a  huge  cavetto  orna- 


PROPORTIONS   OF  THE   ROMAN   CORINTHIAN. 


UNIVERSITY 


FIG.  io.— ROMAN  CORINTHIAN. 


ARCHITECTURE. 


*»«  WMWKM^-Q^WW&AWW^^ 


Fie.   ii.— COMPOSITE  ORDER— ARCH  OF  TITUS. 


MASSIVENESS   OF  EGYPTIAN   STRUCTURES.  33 

mented  with  sculpture ;  and  the  intercolumniation  very  nar- 
row, usually  i£  diameters  and  seldom  exceeding  2^.  In  the 
remains  of  a  temple  the  walls  were  found  to  be  24  feet  thick  ; 
and  at  the  gates  of  Thebes,  the  walls  at  the  foundation  were 
50  feet  thick  and  perfectly  solid.  The  immense  stones  of 
which  these,  as  well  as  Egyptian  walls  generally,  were  built, 
had  both  their  inside  and  outside  surfaces  faced,  and  the 
oints  throughout  the  body  of  the  wall  as  perfectly  close  as 
upon  the  outer  surface.  For  this  reason,  as  well  as  that  the 
buildings  generally  partake  of  the  pyramidal  form,  arise 
their  great  solidity  and  durability.  The  dimensions  and  ex- 
tent of  the  buildings  may  be  judged  from  the  temple  of 
Jupiter  at  Thebes,  which  was  1400  feet  long  and  300  feet 
wide — exclusive  of  the  porticos,  of  which  there  was  a  great 
number. 

It  is  estimated  by  Mr.  Gliddon,  U.  S.  Consul  in  Egypt, 
that  not  less  than  25,000,000  tons  of  hewn  stone  were  em- 
ployed in  the  erection  of  the  Pyramids  of  Memphis  alone — 
or  enough  to  construct  3000  Bunker  Hill  monuments.  Some 
of  the  blocks  are  40  feet  long,  and  polished  with  emery  to  a 
surprising  degree.  It  is  conjectured  that  the  stone  for  these 
pyramids  was  brought,  by  rafts  and  canals,  from  a  distance 
of  six  or  seven  hundred  miles. 

The  general  appearance  of  the  Egyptian  style  of  archi- 
tecture is  that  of  solemn  grandeur — amounting  sometimes  to 
sepulchral  gloom.  For  this  reason  it  is  appropriate  for  cem- 
eteries, prisons,  etc. ;  and  being  adopted  for  these  purposes, 
it  is  gradually  gaining  favor. 

A  great  dissimilarity  exists  in  the  proportion,  form,  and 
general  features  of  Egyptian  columns.  In  some  instances, 
there  is  no  uniformity  even  in  those  of  the  same  building, 
each  differing  from  the  others  either  in  its  shaft  or  capital. 
For  practical  use  in  this  country,  Fig.  13  may  be  taken  as  a 
standard  of  this  style.  The  Halls  of  Justice  in  Centre 
Street,  New  York  City,  is  a  building  in  general  accordance 
with  the  principles  of  Egyptian  architecture. 

46. — Buildings  in  General. — In  selecting  a  style  for  an 
edifice,  its  peculiar  requirements  must  be  allowed  to  govern. 


34 


ARCHITECTURE. 


736* 


in 


ir, 


41 


Fu;.  12. — MODIFIED  TUSCAN  ORDER. 


FITNESS   OF   STYLES.  35 

That  style  of  architecture  is  to  be  preferred  in  which  utility, 
stability,  and  regularity  are  gracefully  blended  with  gran- 
deur and  elegance.  But  as  an  arrangement  designed  for  a 
warm  country  would  be  inappropriate  for  a  colder  climate, 
it  would  seem  that  the  style  of  building  ought  to  be  modified 
to  suit  the  wants  of  the  people  for  whom  it  is.  designed. 
High  roofs  to  resist  the  pressure  of  heavy  snows,  and  ar- 
rangements for  artificial  heat,  are  indispensable  in  northern 
climes ;  while  they  would  be  regarded  as  entirely  out  of 
place  in  buildings  at  the  equator. 

Among  the  Greeks,  architecture  was  employed  chiefly 
upon  their  temples  and  other  large  buildings ;  and  the  pro- 
portions of  the  orders,  as  determined  by  them,  when  execu- 
ted to  such  large  dimensions,  have  the  happiest  effect.  But 
when  used  for  small  buildings,  porticos,  porches,  etc.,  espe- 
cially in  country  places,  they  are  rather  heavy  and  clumsy  ; 
in  such  cases,  more  slender  proportions  will  be  found  to  pro- 
duce a  better  effect.  The  English  cottage-style  is  rather 
more  appropriate,  and  is  becoming  extensively  practised  for 
small  buildings  in  the  country. 

47. — Expression. — Every  building  should  manifest  its 
destination.  If  it  be  intended  for  national  purposes,  it 
should  be  magnificent — grand  ;  for  a  private  residence,  neat 
and  modest ;  for  a  banqueting-house,  gay  and  splendid  ;  for 
a  monument  or  cemetery,  gloomy — melancholy  ;  or,  if  for  a 
church,  majestic  and  graceful — by  some  it  has  been  said, 
"  somewhat  dark  and  gloomy,  as  being  favorable  to  a  devo- 
tional state  of  feeling  ;"  but  such  impressions  can  only  re- 
sult from  a  misapprehension  of  the  nature  of  true  devotion. 
"  Her  ways  are  ways  of  pleasantness,  and  all  her  paths  are 
peace."  The  church  should  rather  be  a  type  of  that 
brighter  world  to  which  it  leads.  Simply  for  purposes  of 
contemplation,  however,  the  glare  of  the  noonday  light 
should  be  excluded,  that  the  worshipper  may,  with  Milton — 

"Love  the  high,  embowed  roof, 
With  antique  pillars  massy  pr£*f, 
And  storied  windows  richlyjj^ght, 
Casting  a  dim,  religious  ligHt." 


ARCHITECTURE. 


H.P. 


PREVALENCE   OF   WOODEN  DWELLINGS.  37 

However  happily  the  several  parts  of  an  edifice  may  be 
disposed,  and  however  pleasing  it  may  appear  as  a  whole, 
yet  much  depends  upon  its  site,  as  also  upon  the  character 
and  style  of  the  structures  in  its  immediate  vicinity,  and  the 
degree  of  cultivation  of  the  adjacent  country.  A  splendid 
country-seat  should  have  the  out-houses  and  fences  in  the 
same  style  with  itself,  the  trees  and  shrubbery  neatly 
trimmed,  and  the  grounds  well  cultivated. 

48. — Durability. — Europeans  express  surprise  that  we 
build  so  much  with  wood.  And  yet,  in  a  new  country, 
where  wood  is  plenty,  that  this  should  be  so  is  no  cause  for 
wonder.  Still  the  practice  should  not  be  encouraged.  Build- 
ings erected  with  brick  or  stone  are  far  preferable  to  those 
of  wood :  they  are  more  durable ;  not  so  liable  to  injury  by 
fire,  nor  to  need  repairs ;  and  will  be  found  in  the  end  quite 
as  economical.  A  wooden  house  is  suitable  for  a  temporary 
residence  only ;  and  those  who  would  bequeath  a  dwelling 
to  their  children  will  endeavor  to  build  with  a  more  dura- 
ble material.  Wooden  cornices  and  gutters,  attached  to 
brick  houses,  are  objectionable — not  only  on  account  of  their 
frail  nature,  but  also  because  they  render  the  building  liable 
to  destruction  by  fire. 

4-9. — Dwelling-Houses :  are  built  of  various  dimensions 
and  styles,  according  to  their  destination ;  and  to  give  de- 
signs and  directions  for  their  erection,  it  is  necessary  to  know 
their  situation  and  object.  A  dwelling  intended  for  a  gar- 
dener would  require  very  different  dimensions  and  arrange- 
ments from  one  intended  for  a  retired  gentleman — with  his 
servants,  horses,  etc. ;  nor  would  a  house  designed  for  the 
city  be  appropriate  for  the  country.  For  city  houses,  ar- 
rangements that  would  be  convenient  for  one  family  might 
be  very  inconvenient  for  two  or  more.  Figs.  14,  15,  16,  17, 
1 8,  and  19  represent  the  icJinograpJiical  projection,  or  ground- 
plan,  of  the  floors  of  an  ordinary  city  house,  designed  to  be 
occupied  by  one  family  only.  Fig.  21  is  an  elevation,  or 
front  view,  of  the  same  house.  All  these  plans  are  drawn  at 
the  same  scale — which  is  that  at  the  bottom  of 


38  ARCHITECTURE. 

Fig.  14  is  a  Plan  of  the  Under-Cellar. 

a,  is  the  coal-vault,  6  by  10  feet. 

b,  is  the  furnace  for  heating  the  house. 
<:,  d,  are  front  and  rear  areas. 

Fig.  15  is  a  Plan  of  the  Basement. 

a,  is  the  library,  or  ordinary  dining-room,  15  by  20  feet. 
by  is  the  kitchen,  15  by  22  feet. 

c,  is  the  store-room,  6  by  9  feet. 

d,  is  the  pantry,  4  by  7  feet. 

e,  is  the  china  closet,  4  by  7  feet. 
fj  is  the  servants'  water-closet. 
g,  is  a  closet. 

//,  is  a  closet  with  a  dumb-waiter  to  the  first  story  above. 
i,  is  an  ash  closet  under  the  front  stoop. 
j,  is  the  kitchen-range. 

k,  is  the  sink  for  washing  and  drawing  water. 
/,  are  wash-trays. 

Fig.  1 6  is  a  Plan  of  the  First  Story. 

a,  is  the  parlor,  1 5  by  34  feet. 

b,  is  the  dining-room,  16  by  23  feet. 

c,  is  the  vestibule. 

<•,  is  the  closet  containing  the  dumb-waiter  from  the  base- 
ment. 

/,  is  the  closet  containing  butler's  sink. 
g,  gy  are  closets. 

//,  is  a  closet  for  hats  and  cloaks. 
iyjy  are  front  and  rear  balconies. 

Fig.  17  is  the  Second  Story. 

a,  <?,  are  chambers,  15  by  13  feet. 

b,  is  a  bed-room,  7^  by  13  feet. 

c,  is  the  bath-room,  7^  by  13  feet. 

d,  dy  are  dressing-rooms,  6  by  7^  feet. 
Cy  e,  are  closets. 

/,  /,  are  wardrobes. 
g,  g,  ve  cupboards. 


PLANS   OF  A  CITY   HOUSE. 


39 


FIG.  14. 

UNDBR-CELLAR. 


FIG-  15- 

BASEMENT. 


CITY  DWELLING. 


ARCHITECTURE. 


CITY  DWELLING. 


FIG.  17. 

SECOND  STORY. 


UPPER   STORIES   OF  A  CITY   HOUSE. 

Fig.  1 8  is  the  Third  Story. 

a,  a,  are  chambers,  15  by  19  feet. 

b,  b,  are  bed-rooms,  ;£  by  13  feet. 

c,  c,  are  closets. 

d,  is  a  linen-closet,  5  by  7  feet. 


FIG.  19. 
FOURTH  STORY. 
CITY  DWELLING. 


e,  e,  are  dressing-closets. 
f,f,  are  wardrobes. 
g,  g,  are  cupboards. 

Fig.  19  is  the  Fourth  Story. 
a,  a,  are  chambers,  14  by  17  feet. 


42  ARCHITECTURE. 

b,  b,  are  bed-rooms,  8£  by  17  feet. 

c,  c,  c,  are  closets. 

d,  is  the  step-ladder  to  the  roof. 

Fig.  20  is  the  Section  of  the  House  showing  the  heights 
of  the  several  stories. 

Fig.  2 1  is  the  Front  Elevation. 

The  size  of  the  house  is  25  feet  front  by  55  feet  deep ;  this 
is  about  the  average  depth,  although  some  are  extended  to 
60  and  65  feet  in  depth. 

These  are  introduced  to  give  some  general  ideas  of  the 
principles  to  be  followed  in  designing  city  houses.  In  plac- 
ing the  chimneys  in  the  parlors,  set  the  chimney-breasts 
equidistant  from  the  ends  of  the  room.  The  basement 
chimney-breasts  may  be  placed  nearly  in  the  middle  of  the 
side  of  the  room,  as  there  is  but  one  flue  to  pass  through 
the  chimney-breast  above  ;  but  in  the  second  story,  as  there 
are  two  flues,  one  from  the  basement  and  one  from  the  par- 
lor, the  breast  will  have  to  be  placed  nearly  perpendicular 
over  the  parlor  breast,  so  as  to  receive  the  flues  within  the 
jambs  of  the  fire-place.  As  it  is  desirable  to  have  the  chim- 
ney-breast as  near  the  middle  of  the  room  as  possible,  it  may 
be  placed  a  few  inches  towards  that  point  from  over  the 
breast  below.  So  in  arranging  those  of  the  stories  above, 
always  make  provision  for  the  flues  from  below. 

50. — Arranging  the  Stairs  and  Window§. — There  should 
be  at  least  as  much  room  in  the  passage  at  the  side  of  the 
stairs  as  upon  them ;  and  in  regard  to  the  length  of  the  pas- 
sage in  the  second  story,  there  must  be  room  for  the  doors 
which  open  from  each  of  the  principal  rooms  into  the  hall, 
and  more  if  the  stairs  require  it.  Having  assigned  a  posi- 
tion for  the  stairs  of  the  second  story,  now  generally  placed 
in  the  centre  of  the  depth  of  the  house,  let  the  winders  of 
the  other  stories  be  placed  perpendicularly  over  and  under 
them ;  and  be  careful  to  provide  for  head-room.  To  ascer- 
tain this,  when  it  is  doubtful,  it  is  well  to  draw  a  vertical 
section  of  the  whole  stairs  ;  but  in  ordinary  cases  this  is  not 


FRONT  OF  A  CITY   HOUSE. 


43 


CITY  DWELLING. 


44  ARCHITECTURE. 

necessary.  To  dispose  the  windows  properly,  the  middle 
window  of  each  story  should  be  exactly  in  the  middle  of  the 
front ;  but  the  pier  between  the  two  windows  which  light 
the  parlor  should  be  in  the  centre  of  that  room  ;  because 
when  chandeliers  or  any  similar  ornaments  hang  from  the 
centre-pieces  of  the  parlor  ceilings,  it  is  important,  in  order 
to  give  the  better  effect,  that  the  pier-glasses  at  the  front 
and  rear  be  in  a  range  with  them.  If  both  these  objects 
cannot  be  attained,  an  approximation  to  each  must  be  at- 
tempted. The  piers  should  in  no  case  be  less  in  width  than 
the  window  openings,  else  the  blinds  or  shutters,  when 
thrown  open,  will  interfere  with  one  another  ;  in  general 
practice,  it  is  well  to  make  the  outside  piers  f  of  the  width 
of  one  of  the  middle  piers.  When  this  is  desirable,  deduct 
the  amount  of  the  three  openings  from  the  width  of  the 
front,  and  the  remainder  will  be  the  amount  of  the  width  of 
all  the  piers  ;  divide  this  by  10,  and  the  product  will  be  ^  of 
a  middle  pier  ;  and  then,  if  the  parlor  arrangements  do  not 
interfere,  give  twice  this  amount  to  each  corner  pier,  and 
three  times  the  same  amount  to  each  of  the  middle  piers. 

51. — Principles  of  Architecture. — To  build  well  requires 
close  attention  and  much  experience.  The  science  of  build- 
ing is  the  result  of  centuries  of  study.  Its  progress  towards 
perfection  must  have  been  exceedingly  slow.  In  the  con- 
struction of  the  first  frail  and  rude  habitations  of  men,  the 
primary  object  was,  doubtless,  utility — a  mere  shelter  from 
sun  and  rain.  But  as  successive  storms  shattered  his  poor 
tenement,  man  was  taught  by  experience  the  necessity  of 
building  with  an  idea  to  durability.  And  as  the  symmetry, 
proportion,  and  beauty  of  nature  met  his  admiring  gaze, 
contrasting  so  strangely  with  the  misshapen  and  dispropor- 
tioned  work  of  his  own  hands,  he  was  led  to  make  gradual 
changes,  till  his  abode  was  rendered  not  only  commodious 
and  durable,  but  pleasant  in  its  appearance  ;  and  building 
became  a  fine  art,  having  utility  for  its  basis. 

52. — Arrangement. — In  all  designs  for  buildings  of  im- 
portance, utility,  durability,  and  beauty,  the  first  great  prin- 
ciples, should  be  pre-eminent.  In  order  that  the  edifice  be 


•  Ox 

UNIVERSITY 


ESSENTIAL   REQUIREMENTS   OF  A   BUILDING.  45 

useful,  commodious,  and  comfortable,  the  arrangement  of  the 
apartments  should  be  such  as  to  fit  them  for  their  several 
destinations  ;  for  publio  assemblies,  oratory,  state,  visitors, 
retiring,  eating,  reading,  sleeping,  bathing,  dressing,  etc.— 
these  should  each  have  its  own  peculiar  form  and  situation. 
To  accomplish  this,  and  at  the  same  time  to  make  their 
relative  situation  agreeable  and  pleasant,  producing  regu- 
larity and  harmony,  require 'in  some  instances  much  skill 
and  sound  judgment.  Convenience  and  regularity  are  very 
important,  and  each  should  have  due  attention  ;  yet  when 
both  cannot  be  obtained,  the  latter  should  in  most  cases 
give  place  to  the  former.  A  building  that  is  neither  con- 
venient nor  regular,  whatever  other  good  qualities  it  may 
possess,  will  be  sure  of  disapprobation. 

53. — Ventilation. — Attention  should  be  given  to  such 
arrangements  as  are  calculated  to  promote  health  :  among 
these,  ventilation  is  by  no  means  the  least.  For  this  pur- 
pose, the  ceilings  of  the  apartments  should  have  a  respect- 
able height ;  and  the  sky-light,  or  any  part  of  the  roof  that 
can  be  made  movable,  should  be  arranged  with  cord  and 
pullies,  so  as  to  be  easily  raised  and  lowered.  Small  open- 
ings near  the  ceiling,  that  may  be  closed  at  pleasure,  should 
be  made  in  the  partitions  that  separate  the  rooms  from  the 
passages  —  especially  for  those  rooms  which  are  used  for 
sleeping  apartments.  All  the  apartments  should  be  so  ar- 
ranged as  to  secure  their  being  easily  kept  dry  and  clean. 
In  dwellings,  suitable  apartments  should  be  fitted  up  for 
bathing  with  all  the  necessary  apparatus  for  conveying 
water. 

54. — Stability. — To  secure  this,  an  edifice  should  be  de- 
signed upon  well-known  geometrical  principles :  such  as 
science  has  demonstrated  to  be  necessary  and  sufficient  for 
firmness  and  durability.  It  is  well,  also,  that  it  have  the 
appearance  of  stability  as  well  as  the  reality ;  for  should  it 
seem  tottering  and  unsafe,  the  sensation  of  fear,  rather  than 
those  of  admiration  and  pleasure,  Avill  be  excited  in  the  be- 
holder. To  secure  certainty  and  accuracy  in  the  applica- 
tion of  those  principles,  a  knowledge  of  the  strength  and 


46  ARCHITECTURE. 

other  properties  of  the  materials  used  is  indispensable  ;  and 
in  order  that  the  whole  design  be  so  made  as  to  be  capable 
of  execution,  a  practical  knowledge  of  the  requisite  mechan- 
ical operations  is  quite  important. 

55. — Decoration. — The  elegance  of  a  design,  although 
chiefly  depending  upon  a  just  proportion  and  harmony  of 
the  parts,  will  be  promoted  by  the  introduction  of  orna- 
ments, provided  this  be  judiciously  performed  ;  for  enrich- 
ments should  not  only  be  ot  a  proper  character  to  suit  the 
style  of  the  building,  but  should  also  have  their  true  posi- 
tion, and  be  bestowed  in  proper  quantity.  The  most  com- 
mon fault,  and  one  which  is  prominent  in  Roman  architec- 
ture, is  an  excess  of  enrichment :  an  error  which  is  carefully 
to  be  guarded  against.  But  those  who  take  the  Grecian 
models  for  their  standard  will  not  be  liable  to  go  to  that 
extreme.  In  ornamenting  a  cornice,  or  any  other  assem- 
blage ol  mouldings,  at  least  every  alternate  member  should 
be  left  plain  ;  and  those  that  are  near  the  eye  should  be  more 
finished  than  those  which  are  distant.  Although  the  charac- 
teristics of  good  architecture  are  utility  and  elegance,  in 
connection  with  durability,  yet  some  buildings  are  designed 
expressly  for  use,  and  others  again  for  ornament :  in  the 
former,  utility,  and  in  the  latter,  beauty,  should  be  the  gov- 
erning principle. 

56. — Elementary  Parts  of  a  Building.  —  The  builder 
should  be  acquainted  with  the  principles  upon  which  the 
essential,  elementary  parts  of  a  building  are  founded.  A 
scientific  knowledge  of  these  will  insure  certainty  and  secu- 
rity, and  enable  the  mechanic  to  erect  the  most  extensive 
and  lofty  edifices  with  confidence.  The  more  important 
parts  are  the  foundation,  the  column,  the  wall,  the  lintel, 
the  arch,  the  vault,  the  dome,  and  the  roof.  A  separate 
description  of  the  peculiarities  of  each  would  seem  to  be 
necessary,  and  cannot  perhaps  be  better  expressed  than  in 
the  following  language  of  a  modern  writer  on  this  subject, 
slightly  modified : 


UNIVERSITY 


STRUCTURAL   FEATURES   OF  A   BUILDING.  47 

57. — The  Foundation:  of  a  building  should  be  begun 
at  a  certain  depth  in  the  earth,  to  secure  a  solid  basis,  below 
the  reach  of  frost  and  common  accidents.  The  most  solid 
basis  is  rock,  or  gravel  which  has  not  been  moved.  Next 
to  these  are  clay  and  sand,  provided  no  other  excavations 
have  been  made  in  the  immediate  neighborhood.  From 
this  basis  a  stone  wall  is  carried  up  to  the  surface  of  the 
ground,  and  constitutes  the  foundation.  Where  it  is  in- 
tended that  the  superstructure  shall  press  unequally,  as  at 
its  piers,  chimneys,  or  columns,  it  is  sometimes  of  use  to 
occupy  the  space  between  the  points  of  pressure  by  an 
inverted  arch.  This  distributes  the  pressure  equally,  and 
prevents  the  foundation  from  springing  between  the  differ- 
ent points.  In  loose  or  muddy  situations,  it  is  always  un- 
safe to  build,  unless  we  can  reach  the  solid  bottom  below. 
In  salt  marshes  and  flats,  this  is  done  by  depositing  timbers, 
or  driving  wooden  piles  into  the  earth,  and  raising  walls 
upon  them.  The  preservative  quality  of  the  salt  will  keep 
these  timbers  unimpaired  for  a  great  length  of  time,  and 
makes  the  foundation  equally  secure  with  one  of  brick  or 
stone. 

58. — The  Column,  or  Pillar:  is  the  simplest  member  in 
any  building,  though  by  no  means  an  essential  one  to  all. 
This  is  a  perpendicular  part,  commonly  of  equal  breadth 
and  thickness,  not  intended  for  the  purpose  of  enclosure, 
but  simply  for  the  support  of  some  part  of  the  superstruc- 
ture. The  principal  force  which  a  column  has  to  resist  is 
that  of  perpendicular  pressure.  In  its  shape,  the  shaft  of  a 
column  should  not  be  exactly  cylindrical,  but,  since  the 
lower  part  must  support  the  weight  of  the  superior  part,  in 
addition  to  the  weight  which  presses  equally  on  the  whole 
column,  the  thickness  should  gradually  decrease  from  bot- 
tom to  top.  The  outline  of  columns  should  be  a  little 
curved,  so  as  to  represent  a  portion  of  a  very  long  spheroid, 
or  paraboloid,  rather  than  of  a  cone.  This  figure  is  the  joint 
result  of  two  calculations,  independent  of  beauty  of  appear- 
ance. One  of  these  is,  that  the  form  best  adapted  for  sta- 
bility of  base  is  that  of  a  cone  ;  the  other  is,  that  the  figure, 


48  ARCHITECTURE. 

which  would  be  of  equal  strength  throughout  for  support- 
ing a  superincumbent  weight,  would  be  generated  by  the 
revolution  of  two  parabolas  round  the  axis  of  the  column, 
the  vertices  of  the  curves  being  at  its  extremities.  The 
swell  of  the  shafts  of  columns  was  called  the  entasis  by  the 
ancients.  It  has  been  lately  found  that  the  columns  of  the 
Parthenon,  at  Athens,  which  have  been  commonly  supposed 
straight,  deviate  about  an  inch  from  a  straight  line,  and  that 
their  greatest  swell  is  at  about  one  third  of  their  height. 
Columns  in  the  antique  orders  are  usually  made  to  diminish 
one  sixth  or  one  seventh  of  their  diameter,  and  sometimes 
even  one  fourth.  The  Gothic  pillar  is  commonly  of  equal 
thickness  throughout. 

59. — The  Wall :  another  elementary  part  of  a  building, 
may  be  considered  as  the  lateral  continuation  of  the  column, 
answering  the  purpose  both  of  enclosure  and  support.  A 
wall  must  diminish  as  it  rises,  for  the  same  reasons,  and  in 
the  same  proportion,  as  the  column.  It  must  diminish  still 
more  rapidly  if  it  extends  through  several  stories,  support- 
ing weights  at  different  heights.  A  wall,  to  possess  the 
greatest  strength,  must  also  consist  of  pieces,  the  upper  and 
lower  surfaces  of  which  are  horizontal  and  regular,  not 
rounded  nor  oblique.  The  walls  of  most  of  the  ancient 
structures  which  have  stood  to  the  present  time  are  con- 
structed in  this  manner,  and  frequently  have  their  stones 
bound  together  with  bolts  and  clamps  of  iron.  The  same 
method  is  adopted  in  such  modern  structures  as  are  intended 
to  possess  great  strength  and  durability,  and,  in  some  cases, 
the  stones  are  even  dovetailed  together,  as  in  the  light- 
houses at  Eddy  stone  and  Bell  Rock.  But  many  of  our 
modern  stone  walls,  for  the  sake  of  cheapness,  have  only  one 
face  of  the  stones  squared,  the  inner  half  of  the  wall  being 
completed  with  brick ;  so  that  they  can,  in  reality,  be  con- 
sidered only  as  brick  walls  faced  with  stone.  Such  walls  are 
said  to  be  liable  to  become  convex  outwardly,  from  the  dif- 
ference in  the  shrinking  of  the  cement.  Rubble  walls  are 
made  of  rough,  irregular  stones,  laid  in  mortar.  The  stones 
should  be  broken,  if  possible,  so  as  to  produce  horizontal 


VARIOUS    METHODS   OF   ERECTING   WALLS.  49 

surfaces.  The  coffer  walls  of  the  ancient  Romans  were  made 
by  enclosing  successive  portions  of  the  intended  wall  in  a 
box,  and  filling  it  with  stones,  sand,  and  mortar  promis- 
cuously. This  kind  of  structure  must  have  been  extremely 
insecure.  The  Pantheon  and  various  other  Roman  build- 
ings are  surrounded  with  a  double  brick  wall,  having  its 
vacancy  filled  up  with  loose  bricks  and  cement.  The  whole 
has  gradually  consolidated  into  a  mass  of  great  firmness. 

60. — The  Reticulated  Walls  :  of  the  Romans  —  com- 
posed of  bricks  with  oblique  surfaces — would,  at  the  present 
day,  be  thought  highly  unphilosophical.  Indeed,  they  could 
not  long  have  stood,  had  it  not  been  for  the  great  strength 
of  their  cement.  Modern  brick  walls  are  laid  with  great 
precision,  and  depend  for  firmness  more  upon  their  position 
than  upon  the  strength  of  their  cement.  The  bricks  being 
laid  in  horizontal  courses,  and  continually  overlaying  each 
other,  or  breaking  joints,  the  whole  mass  is  strongly  inter- 
woven, and  bound  together.  Wooden  walls,  composed  of 
timbers  covered  with  boards,  are  a  common  but  more  per- 
ishable kind.  They  require  to  be  constantly  covered  with  a 
coating  of  a  foreign  substance,  as  paint  or  plaster,  to  pre- 
serve them  from  spontaneous  decomposition.  In  some  parts 
of  France,  and  elsewhere,  a  kind  of  wall  is  made  of  earth, 
rendered  compact  by  ramming  it  in  moulds  or  cases.  This 
method  is  called  building  in  pise,  and  is  much  more  durable 
than  the  nature  of  the  material  would  lead  us  to  suppose. 
Walls  of  all  kinds  are  greatly  strengthened  by  angles  and 
curves,  also  by  projections,  such  as  pilasters,  chimneys,  and 
buttresses.  These  projections  serve  to  increase  the  breadth 
of  the  foundation,  and  are  always  to  be  made  use  of  in  large 
buildings,  and  in  walls  of  considerable  length. 

61. — The  Lintel,  or  Beam:  extends  in  a  right  line  over 
a  vacant  space,  from  one  column  or  wall  to  another.  The 
strength  of  the  lintel  will  be  greater  in  proportion  as  its 
transverse  vertical  diameter  exceeds  the  horizontal,  the 
strength  being  always  as  the  square  of  the  depth.  The 
floor  is  the  lateral  continuation  or  connection  of  beams  by 
means  of  a  covering  of  boards. 


50  ARCHITECTURE. 

62.— The  Arch :  is  a  transverse  member  of  a  building, 
answering  the  same  purpose  as  the  lintel,  but  vastly  exceed- 
ing it  in  strength.  The  arch,  unlike  the  lintel,  may  consist 
of  any  number  of  constituent  pieces,  without  impairing  its 
strength.  It  is,  however,  necessary  that  all  the  pieces  should 
possess  a  uniform  shape, —  the  shape  of  a  portion  of  a 
wedge, — and  that  the  joints,  formed  by  the  contact  of  their 
surfaces,  should  point  towards  a  common  centre.  In  this 
case,  no  one  portion  of  the  arch  can  be  displaced  or  forced 
inward  ;  and  the  arch  cannot  be  broken  by  any  force  which 
is  not  sufficient  to  crush  the  materials  of  which  it  is  made. 
In  arches  made  of  common  bricks,  the  sides  of  which  are 
parallel,  any  one  of  the  bricks  might  be  forced  inward,  were 
it  not  for  the  adhesion  of  the  cement.  Any  two  of  the  bricks, 
however,  by  the  disposition  of  their  mortar,  cannot  collect- 
ively be  forced  inward.  An  arch  of  the  proper  form,  when 
complete,  is  rendered  stronger,  instead  of  weaker,  by  the 
pressure  of  a  considerable  weight,  provided  this  pressure  be 
uniform.  While  building,  however,  it  requires  to  be  sup- 
ported by  a  centring  of  the  shape  of  its  internal  surface, 
until  it  is  complete.  The  upper  stone  of  an  arch  is  called 
the  keystone,  but  is  not  more  essential  than  any  other.  In 
regard  to  the  shape  of  the  arch,  its  most  simple  form  is  that 
of  the  semicircle.  It  is,  however,  very  frequently  a  smaller 
arc  of  a  circle,  or  a  portion  of  an  ellipse. 

63. — Hooke'§  Theory  of  an  Arch. — The  simplest  theory 
of  an  arch  supporting  itself  only  is  that  of  Dr.  Hooke. 
The  arch,  when  it  has  only  its  own  weight  to  bear,  may  be 
considered  as  the  inversion  of  a  chain,  suspended  at  each 
end.  The  chain  hangs  in  such  a  form  that  the  weight  of 
each  link  or  portion  is  held  in  equilibrium  by  the  result  of 
two  forces  acting  at  its  extremities  ;  and  these  forces,  or 
tensions,  are  produced,  the  one  by  the  weight  of  the  portion 
of  the  chain  below  the  link,  the  other  by  the  same  weight 
increased  by  that  of  the  link  itself,  both  of  them  acting  ori- 
ginally in  a  vertical  direction.  Now,  supposing  the  chain 
inverted,  so  as  to  constitute  an  arch  of  the  same  form  and 
weight,  the  relative  situations  of  the  forces  will  be  the  same, 


VIADUCT   AT   CHAUMONT. 


PECULIARITIES   OF  THE  ARCH.  51 

only  they  will  act  in  contrary  directions,  so  that  they  are 
compounded  in  a  similar  manner,  and  balance  each  other  on 
the  same  conditions. 

The  arch  thus  formed  is  denominated  a  catenary  arch. 
In  common  cases,  it  differs  but  little  from  a  circular  arch  of 
the  extent  of  about  one  third  of  a  whole  circle,  and  rising 
from  the  abutments  with  an  obliquity  of  about  30  degrees 
from  a  perpendicular.  But  though  the  catenary  arch  is  the 
best  form  for  supporting  its  own  weight,  and  also  all  addi- 
tional weight  which  presses  in  a  vertical  direction,  it  is  not 
the  best  form  to  resist  lateral  pressure,  or  pressure  like  that 
of  fluids,  acting  equally  in  all  directions.  Thus  the  arches 
of  bridges  and  similar  structures,  when  covered  with  loose 
stones  and  earth,  are  pressed  sideways,  as  well  as  vertically, 
in  the  same  manner  as  if  they  supported  a  weight  of  fluid. 
In  this  case,  it  is  necessary  that  the  arch  should  arise  more 
perpendicularly  from  the  abutment,  and  that  its  general  fig- 
ure should  be  that  of  the  longitudinal  segment  of  an  ellipse. 
In  small  arches,  in  common  buildings,  where  the  disturbing 
force  is  not  great,  it  is  of  little  consequence  what  is  the 
shape  of  the  curve.  The  outlines  may  even  be  perfectly 
straight,  as  in  the  tier  of  bricks  which  we  frequently  see 
over  a  window.  This  is,  strictly  speaking,  a  real  arch,  pro- 
vided the  surfaces  of  the  bricks  tend  toward  a  common 
centre.  It  is  the  weakest  kind  of  arch,  and  a  part  of  it 
is  necessarily  superfluous,  since  no  greater  portion  can  act 
in  supporting  a  weight  above  it  than  can  be  included  be- 
tween two  curved  or  arched  lines. 

f 

64.  —  Gothic  Arches.  —  Besides  these  arches,  various 
others  are  in  use.  The  acute  or  lancet  arch,  much  used  in 
Gothic  architecture,  is  described  usually  from  two  cen- 
tres outside  the  arch.  It  is  a  strong  arch  for  supporting 
vertical  pressure.  The  rampant  arch  is  one  in  which  the  two 
ends  spring  from  unequal  heights.  The  Jwrseshoe  or  Moorisli 
arch  is  described  from  one  or  more 'centres  placed  above  the 
base  line.  In  this  arch,  the  lower  parts  are  in  danger  of 
being  forced  inward.  The  ogee  arch  is  concavo-convex,  and 
therefore  fit  only  for  ornament. 


52  ARCHITECTURE. 

65. — Arch:  Definition*;  Principles.  —  The  upper  sur- 
face is  called  the  extrados,  and  the  inner,  the  intrados. 
The  spring  is  where  the  intrados  meets  the  abutments.  The 
span  is  the  distance  between  the  abutments.  The  wedge- 
shaped  stones  which  form  an  arch  are  sometimes  called 
vonssoirs,  the  uppermost  being-  the  keystone.  The  part  of  a 
pier  from  which  an  arch  springs  is  called  the  impost,  and  the 
curve  formed  by  the  under  side  of  the  voussoirs,  the  archi- 
volt.  It  is  necessary  that  the  walls,  abutments,  and  piers  on 
which  arches  are  supported  should  be  so  firm  as  to  resist  the 
lateral  thrust,  as  well  as  vertical  pressure,  of  the  arch. 
It  will  at  once  be  seen  that  the  lateral  or  sideway  pressure 
of  an  arch  is  very  considerable,  when  we  recollect  that  every 
stone,  or  portion  of  the  arch,  is  a  wedge,  a  part  of  whose 
force  acts  to  separate  the  abutments.  For  want  of  attention 
to  this  circumstance,  important  mistakes  have  been  commit- 
ted, the  strength  of  buildings  materially  impaired,  and  their 
ruin  accelerated.  In  some  cases,  the  want  of  lateral  firmness 
in  the  walls  is  compensated  by  a  bar  of  iron  stretched  across 
the  span  of  the  arch,  and  connecting  the  abutments,  like  the 
tie-beam  of  a  roof.  This  is  the  case  in  the  cathedral  of  Milan 
and  some  other  Gothic  buildings. 

66. — An  Arcade :  or  continuation  of  arches,  needs  only 
that  the  outer  supports  of  the  terminal  arches  should  be 
strong  enough  to  resist  horizontal  pressure.  In  the  inter- 
mediate arches,  the  lateral  force  of  each  arch  is  counter- 
acted by  the  opposing  lateral  force  of  the  one  contiguous  to 
it.  In  bridges,  however,  where  individual  arches  are  liable 
to  be  destroyed  by  accident,  it  is  desirable  that  each  of  the 
piers  should  possess  sufficient  horizontal  strength  to  resist 
the  lateral  pressure  of  the  adjoining  arches. 

67. — The  Vault:  is  the  lateral  continuation  of  an  arch, 
serving  to  cover  an  area  or  passage,  and  bearing  the  same 
relation  to  the  arch  that  the  wall  does  to  the  column.  A 
simple  vault  is  constructed  on  the  principles  of  the  arch,  and 
distributes  its  pressure  equally  along  the  walls  or  abutments. 
A  complex  or  groined  vault  is  made  by  two  vaults  intersect^ 
ing  each  other,  in  which  case  the  pressure  is  thrown  upon 


VARIOUS   CONSTRUCTIONS   OF  THE   DOME.  53 

springing  points,  and  is  greatly  increased  at  those  points. 
The  groined  vault  is  common  in  Gothic  architecture. 

68. — The  Dome:  sometimes  called  cupola,'^  a  concave 
covering  to  a  building,  or  part  of  it,  and  may  be  either  a 
segment  of  a  sphere,  of  a  spheroid,  or  of  any  similar  figure. 
When  built  of  stone,  it  is  a  very  strong  kind  of  structure, 
even  more  so  than  the  arch,  since  the  tendency  of  each  part 
to  fall  is  counteracted,  not  only  by  those  above  and  below  it, 
but  also  by  those  on  each  side.  It  is  only  necessary  that 
the  constituent  pieces  should  have  a  common  form,  and  that 
this  form  should  be  somewhat  like  the  frustum  of  a  pyra- 
mid, so  that,  when  placed  in  its  situation,  its  four  angles  may 
point  toward  the  centre,  or  axis,  of  the  dome.  During  the 
erection  of  a  dome,  it  is  not  necessary  that  it  should  be  sup- 
ported by  a  centring,  until  complete,  as  is  done  in  the  arch. 
Each  circle  of  stones,  when  laid,  is  capable  of  supporting 
itself  without  aid  from  those  above  it.  It  follows  that  the 
dome  may  be  left  open  at  top,  without  a  keystone,  and 
yet  be  perfectly  secure  in  this  respect,  being  the  reverse  of 
the  arch.  The  dome  of  the  Pantheon,  at  Rome,  has  been 
always  open  at  top,  and  yet  has  stood  unimpaired  for  nearly 
2006  years.  The  upper  circle  of  stones,  though  apparently 
the  weakest,  is  nevertheless  often  made  to  support  the  addi- 
tional weight  of  a  lantern  or  tower  above  it.  In  several  of 
the  largest  cathedrals,  there  are  two  domes,  one  within  the 
other,  which  contribute  their  joint  support  to  the  lantern, 
which  rests  upon  the  top.  In  these  buildings,  the  dome 
rests  upon  a  circular  wall,  which  is  supported,  in  its  turn,  by 
arches  upon  massive  pillars  or  piers.  This  construction  is 
called  building  upon  pendentivcs,  and  gives  open  space  and 
room  for  passage  beneath  the  dome.'  The  remarks  which 
have  been  made  in  regard  to  the  abutments  of  the  arch 
apply  equally  to  the  walls  immediately  supporting  a  dome. 
They  must  be  of  sufficient  thickness  and  solidity  to  resist 
the  lateral  pressure  of  the  dome,  which  is  very  great.  The 
walls  of  the  Roman  Pantheon  are  of  great  depth  and  solid- 
ity. In  order  that  a  dome  in  itself  should  be  perfectly 
secure,  its  lower  parts  must  not  be  too  nearly  vertical,  since, 


54  ARCHITECTURE. 

in  this  case,  they-  partake  of  the  nature  of  perpendicular 
walls,  and  are  acted  upon  by  the  spreading  force  of  the 
parts  above  them.  The  dome  of  St.  Paul's  Church,  in  Lon- 
don, and  some  others  of  similar  construction,  are  bound  with 
chains  or  hoops  of  iron,  to  prevent  them  from  spreading  at 
bottom.  Domes  which  are  made  of  wood  depend,  in  part* 
for  their  strength  on  their  internal  carpentry.  The  Halle 
du  Bled,  in  Paris,  had  originally  a  wooden  dome  more  than 
200  feet  in  diameter,  and  only  one  foot  in  thickness.  This 
has  since  been  replaced  by  a  dome  of  iron.  (See  Art- 

235.) 

69. — Tlie  Roof:  is  the  most  common  and  cheap  method 
of  covering  buildings,  to  protect  them  from  rain  and  other 
effects  of  the  weather.  It  is  sometimes  flat,  but  more  fre- 
quently oblique,  in  its  shape.  The  flat  or  platform  roof  is 
the  least  advantageous  for  shedding  rain,  and  is  seldom  used 
in  northern  countries.  The  pent  roof,  consisting  of  two 
oblique  sides  meeting  at  top,  is  the  most  common  form. 
These  roofs  are  made  steepest  in  cold  climates,  where  they 
are  liable  to  be  loaded  with  snow.  Where  the  four  sides  of 
the  roof  are  all  oblique,  it  is  denominated  a  hipped  roof,  and 
where  there  are  two  portions  to  the  roof,  of  different  ob- 
liquity, it  is  a  curb,  or  mansard  roof.  In  modern  times,  roofs 
are  made  almost  exclusively  of  wood,  though  frequently 
covered  with  incombustible  materials.  The  internal  struc- 
ture or  carpentry  of  roofs  is  a  subject  of  considerable  me- 
chanical contrivance.  The  roof  is  supported  by  rafters, 
which  abut  on  the  walls  on  each  side,  like  the  extremities  of 
an  arch.  If  no  other  timbers  existed  except  the  rafters, 
they  would  exert  a  strong  lateral  pressure  on  the  walls, 
tending  to  separate  and  overthrow  them.  To  counteract 
this  lateral  force,  a  tic-beam,  as  it  is  called,  extends  across, 
receiving  the  ends  of  the  rafters,  and  protecting  the  wall 
from  their  horizontal  thrust.  To  prevent  the  tie-beam  from 
sagging,  or  bending  downward  with  its  own  weight,  a  king- 
post is  erected  from  this  beam,  to  the  upper  angle  of  the 
rafters,  serving  to  connect  the  whole,  and  to  suspend  the 
weight  of  the  beam.  This  is  called  trussing.  Queen-posts 


MANNER   OF   CONSTRUCTING   ROOFS.  55 

are  sometimes  added,  parallel  to  the  king-post,  in  large  roofs  I 
also  various  other  connecting  timbers.  In  Gothic  buildings, 
where  the  vaults  do  not  admit  of  the  use  of  a  tie-beam,  the 
rafters  are  prevented  from  spreading,  as  in  an  arch,  by  the 
strength  of  the  buttresses. 

In  comparing  the  lateral  pressure  of  a  high  roof  with 
that  of  a  low  one,  the  length  of  the  tie-beam  being  the  same, 
it  will  be  seen  that  a  high  roof,  from  its  containing  most 
materials,  may  produce  the  greatest  pressure,  as  far  as 
weight  is  concerned.  On  the  other  hand,  if  the  weight  of 
both  be  equal,  then  the  low  roof  will  exert  the  greater  press- 
ure ;  and  this  will  increase  in  proportion  to  the  distance  of 
the  point  at  which  perpendiculars,  drawn  from  the  end  of 
each  rafter,  would  meet.  In  roofs,  as  well  as  in  wooden 
domes  and  bridges,  the  materials  are  subjected  to  an  inter- 
nal strain,  to  resist  which  the  cohesive  strength  of  the  ma- 
terial is  relied  on.  On  this  account,  beams  should,  when 
possible,  be  of  one  piece.  Where  this  cannot  be  effected, 
two  or  more  beams  are  connected  together  by  splicing. 
Spliced  beams  are  never  so  strong  as  whole  ones,  yet  they 
may  be  made  to  approach  the  same  strength,  by  affixing  lat- 
eral pieces,  or  by  making  the  ends  overlay  each  other,  and 
connecting  them  with  bolts  and  straps  of  iron.  The  ten- 
dency to  separate  is  also  resisted,  by  letting  the  two  pieces 
into  each  other  by  the  process  called  scarfing.  Mortices,  in- 
tended to  truss  or  suspend  one  piece  by  another,  should  be 
formed  upon  similar  principles. 

Roofs  in  the  United  States,  after  being  boarded,  receive 
a  secondary  covering  of  shingles.  When  intended  to  be 
incombustible,  they  are  covered  with  slates  or  earthen  tiles, 
or  with  sheets  of  lead,  copper,  or  tinned  iron.  Slates  are 
preferable  to  tiles,  being  lighter,  and  absorbing  less  moisture. 
Metallic  sheets  are  chiefly  used  for  flat  roofs,  wooden  domes, 
and  curved  and  angular  surfaces,  which  require  a  flexible 
material  to  cover  them,  or  have  not  a  sufficient  pitch  to  shed 
the  rain  from  slates  or  shingles.  Various  artificial  composi- 
tions are  occasionally  used  to  cover  roofs,  the  most  common 
of  which  are  mixtures  of  tar  with  lime,  and  sometimes  with 
sand  and  gravel. — Ency.  Am.  (See  Art.  202.) 


SECTION  II.— CONSTRUCTION. 

ART.  70. — Con§tructioii  E§seiUiai. — Construction  is  that 
part  of  the  Science  of  Building  which  treats  of  the  Laws  of 
Pressure  and  the  strength  of  materials.  To  the  architect 
and  builder  a  knowledge  of  it  is  absolutely  essential.  It  de- 
serves a  larger  place  in  a  volume  of  this  kind  than  is  gene- 
rally allotted  to  it.  Something,  indeed,  has  been  said  upon 
the  styles  and  principles,  by  which  the  best  arrangements 
may  be  ascertained ;  yet,  besides  this,  there  is  much  to  be 
learned.  For  however  precise  or  workmanlike  the  several 
parts  may  be  made,  what  will  it  avail,  should  the  system  of 
framing,  from  deficient  material,  or  an  erroneous  position  of 
its  timbers,  fail  to  sustain  even  its  own  weight  ?  Hence  the 
necessity  for  a  knowledge  of  the  laws  of  pressure  and  the 
strength  of  materials.  These  being  once  understood,  we 
can  with  confidence  determine  the  best  position  and  dimen- 
sions for  the  several  pieces  which  compose  a  floor  or  a  roof, 
a  partition  or  a  bridge.  As  systems  of  framing  are  more  or 
less  exposed  to  heavy  weights  and  strains,  and,  in  case  of 
failure,  cause  not  only  a  loss  of  labor  and  material,  but  fre- 
quently that  of  life  itself,  it  is  vety  important  that  the  mate- 
rials employed  be  of  the  proper  quantity  and  quality  to  serve 
their  destination.  And,  on  the  other  hand,  any  superfluous 
material  is  not  only  useless,  but  a  positive  injury,  as  it  is  an 
unnecessary  load  upon  the  points  of  support.  It  is  neces- 
sary, therefore,  to  know  the  least  quantity  of  material  that 
will  suffice  for  strength.  Not  the  least  common  fault  in 
framing  is  that  of  using  an  excess  of  material.  Economy,  at 
least,  would  seem  to  require  that  this  evil  be  abated.  < 

Before  proceeding  to  consider  the  principles  upon  which 
a  system  of  framing  should  be  constructed,  let  us  attend  to 
a  few  of  the  elementary  laws  in  Mechanics,  which  will  be 
found  to  be  of  great  value  in  determining  those  principles. 


INTERIOR   OF  THE  CATHEDRAL,    SIENNA. 


DIRECT   AND    OBLIQUE   SUPPORTS. 


57 


71. — ILaws  of  Pre§§ure. — (i.)  A  heavy  body  always  ex- 
erts a  pressure  equal  to  its  own  weight  in  a  vertical  direc- 
tion. Example  :  Suppose  an  iron  ball  weighing  100  Ibs.  be 
supported  upon  the  top  of  a  perpendicular  post  (Fig.  22-A) ; 
then  the  pressure  exerted  upon  that  post  will  be  equal  to 
the  weight  of  the  ball,  viz.,  100  Ibs.  (2.)  But  if  two  inclined 
posts  (Fig.  22-B)  be  substituted  for  the  perpendicular  sup- 
port, the  united  pressures  upon  these  posts  will  be  more 
than  equal  to  the  weight,  and  will  be  in  proportion  to  their 
position.  The  farther  apart  their  feet  are  spread  the  greater 
will  be  the  pressure,  and  vice  versa.  Hence  tremendous 
strains  may  be  exerted  by  a  comparatively  small  weight. 
And  it  follows,  therefore,  that  a  piece  of  timber  intended 
for  a  strut  or  post  should  be  so  placed  that  its  axis  may 
coincide,  as  nearly  as  possible,  with  the  direction  of  the 
pressure.  The  direction  of  the  pressure  of  the  weight  W 
(Fig.  22-B)  is  in  the  vertical  line  b  d\  and  the  weight  W 
would  fall  in  that  line  if  the  two  posts  were  removed  ;  hence 
the  best  position  for  a  support  for  the  weight  would  be  in 


A. 


FlG.    22. 


that  line.  But  as  it  rarely  occurs  in  systems  of  framing 
that  weights  can  be  supported  by  any  single  resistance,  they 
requiring  generally  two  or  more  supports  (as  in  the  case  of 
a  roof  supported  by  its  rafters),  it  becomes  important,  there- 
fore, to  know  the  exact  amount  of  pressure  any  certain 
weight  is  capable  of  exerting  upon  oblique  supports.  Now, 
it  has  been  ascertained  that  the  three  lines  of  a  triangle, 
drawn  parallel  with  the  direction  of  three  concurring  forces 
in  equilibrium,  are  in  proportion  respectively  to  these 


58  CONSTRUCTION. 

forces.  For  example,  in  Fig.  22-B,  we  have  a  representation 
of  three  forces  concurring  in  a  point,  which  forces  are  in 
equilibrium  and  at  rest ;  thus,  the  weight  W  is  one  force, 
and  the  resistances  exerted  by  the  two  pieces  of  timber  are 
the  other  two  forces.  The  direction  in  which  the  first  force 
acts  is  vertical — downwards ;  the  direction  of  the  two  other 
forces  is  in  the  axis  of  each  piece  of  timber  respectively. 
These  three  forces  all  tend  towards  the  point  b. 

Draw  the  axes  a  b  and  b  c  of  the  two  supports ;  make  b  d 
vertical,  and  from  d  draw  d  e  and  d  f  parallel  with  the  axes 
b  c  and  b  a  repectively.  Then  the  triangle  b  d  e  has  its 
lines  parallel  respectively  with  the  direction  of  the  three 
forces ;  thus,  bd  is  in  the  direction  of  the  weight  W,  d  e  paral- 
lel with  the  axis  of  the  timber  D,  and  e  b  is  in  the  direction  of 
the  timber  C.  In  accordance  with  the  principle  above  stated, 
the  lengths  of  the  sides  of  the  triangle  b  d  e  are  in  propor- 
tion respectively  to  the  three  forces  aforesaid  ;  thus — 

As  the  length  of  the  line  b  d 

Is  to  the  number  of  pounds  in  the  weight  W, 

So  is  the  length  of  the  line  £  e 

To  the  number  of  pounds'  pressure  resisted  by  the 

timber  C. 
Again — 

As  the  length  of  the  line  b  d 

Is  to  the  number  of  pounds  in  the  weight  W, 

So  is  the  length  of  the  line  d  c 

To  the  number  of  pounds'  pressure  resisted  by  the 

timber  D. 
And  again — 

As  the  length  of  the  line  b  e 

Is  to  the  pounds'  pressure  resisted  by  C, 

So  is  the  length  of  the  line  d  e 

To  the  pounds'  pressure  resisted  by  D. 

These  proportions  are  more  briefly  stated  thus — 
\st.  bd\  W\\be\P, 

P  being  used  as  a  symbol  to  represent  the  number  of  pounds' 
pressure  resisted  by  the  timber  C. 


PARALLELOGRAM   OF   FORCES.  59 

2d.  b  d  :  W  :  :  d  e  :  Q, 

Q  representing  the  number  of  pounds'  pressure  resisted  by 
the  timber  D. 

3d.  b  e  \P\\de\  Q. 

72. — Parallelogram  of  Forces. — This  relation  between 
lines  and  pressures  is  applicable  in  ascertaining  the  pres- 
sures induced  by  known  weights  throughout  any  system  of 
framing.  The  parallelogram  b  e  df  is  called  the  Parallelo- 
%ram  of  Forces ;  the  two  lines  If  e- and  bf  being  called  the 
components,  and  the  line  b  d  the  resultant.  Where  it  is  re- 
quired to  find  the  components  from  a  given  resultant  (Fig. 
22- B\  the  fourth  line  df/need  not  be  drawn,  for  the  triangle 
b  d  e  gives  the  desired  result.  But  when  the  resultant  is  to 
be  ascertained  from  given  components  (Fig.  28),  it  is  more 
convenient  to  draw  the  fourth  line. 

73. — The  Resolution  of  Forces :  Is  the  finding  of  two 
or  more  forces  which,  acting  in  different  directions,  shall 
exactly  balance  the  pressure  of  any  given  single  force.  To 
make  a  practical  application  of  this,  let  it  be  required  to 
ascertain  the  oblique  pressure  in  Fig.  22-B.  In  this  figure  the 
line  bd  measures  half  an  inch  (0-5  inch),  and  the  line  be 
three  tenths  of  an  inch  (0-3  inch).  Now  if  the  weight  W 
be  supposed  to  be  1200  pounds,  then  the  first  stated  propor- 
tion above, 

b  d\  W: :  b  e  :  P,   becomes   0-5  :  1200  : :  0-3  :  P. 

And  since  the  product  of  the  means  divided  by  one  of  the 
extremes  gives  the  other  extreme,  this  proportion  may  be 
put  in  the  form  of  an  equation^  thus — 

I200X  0-3  _p 

0-5 

Performing  the  arithmetical  operation  here  indicated— -that 
is,  multiplying  together  the  two  quantities  above  the  line, 
and  dividing  the  product  by  the  quantity  under  the  line— the 


6o 


CONSTRUCTION. 


quotient  will  be  equal  to  the  quantity  represented  by  P,  viz., 
the  pressure  resisted  by  the  timber  C.     Thus — 


1 200 


0-5)360-0 


720  =  P. 

The  strain  upon  the  timber  C  is,  therefore,  equal  to  720 
pounds;  and  since,  in  this  case  (the  two  timbers  being  in- 
clined equally  from  the  vertical),  the  line  e  d  is  equal  to  the 
line  b  e,  therefore  the  strain  upon  the  other  timber  D  is 
also  720  pounds. 


FIG.  23. 

74. — Inclination  of  Supports  Unequal. — In  Fig.  23  the 
pressures  in  the  two  supports  are  unequal.  The  supports 
are  also  unequal  in  length.  The  length  of  the  supports, 
however,  does  not  alter  the  amount  of  pressure  from  the 
concentrated  load  supported ;  but  generally  long  timbers 
are  not  so  capable  of  resistance  as  shorter  ones.  For,  not 
being  so  stiff,  they  bend  more  readily,  and,  since  the  com- 
pression is  in  proportion  to  the  length,  they  therefore 
shorten  more.  To  ascertain  the  pressures  in  Fig.  23,  let  the 
weight  suspended  from  b  d  be  equal  to  two  and  three  quarter 
tons  (2-75  tons).  The  line  b  d  measures  five  and  a  half 
tenths  of  an  inch  (0-55  inch),  and  the  line  b  e  half  an  inch 
(0-5  inch).  Therefore,  the  proportion 

b  d\  W \\  b  e  :  P  becomes  0-55  :  2-75  : :  0-5  :  P, 


STRAIN   IN   PROPORTION   TO    INCLINATION.  — £r 

and  2.75x0.5  / 

0-55        (UNIVERSITY 


0'55)i -375(2- 5  =  ^. 

I    10 

275 

275 

The  strain  upon  the  timber  A.  is,  therefore,  equal  to  two 
and  a  half  tons. 

Again,  the  line  e  d  measures  four  tenths  of  an  inch  (0-4 
inch)  ;  therefore,  the  proportion 

b  d  \  W \ :  e  d  \  Q  becomes  0-55  :  2-75  : :  0-4  :  Q, 

and  2./5  x  0-4  __  ^ 

o-55 

2-75 
0-4 

0-55)  1-100(2  =  Q. 

I    10 


The  strain  upon  the  timber  B  is,  therefore,  equal  to  two 
tons. 

'/5. — The  Strains  Exceed  tBie  Weights. — Thus  the  united 
pressures  upon  the  two  inclined  supports  always  exceed  the 
weight.  In  the  last  case,  2f  tons  exert  a  pressure  of  2\  and 
2  tons,  equal  together  to  4^  tons  ;  and  in  the  former  case, 
1200  pounds  exert  a  pressure  of  twice  720  pounds,  equal  to 
1440  pounds.  The  smaller  the  angle  of  inclination  to  the  ' 
horizontal,  the  greater  will  be  the  pressure  upon  the  sup- 
ports. So,  in  the  frame  of  a  roof,  the  strain  upon  the  rafters 
decreases  gradually  with  the  increase  of  the  angle  of  incli- 
nation to  the  horizon,  the  length  of  the  rafter  remaining  the 


same. 


62  CONSTRUCTION. 

This  is  true  in  comparing  one  system  of  framing  with 
another  ;  but  in  a  system  where  the  concentrated  weight  to 
be  supported  is  not  in  the  middle  (see  Fig.  23),  and,  in  con- 
sequence, the  supports  are  not  inclined  equally,  the  strain 
will  be  greatest  upon  that  support  which  has  the  greatest 
inclination  to  the  horizon. 

76. — Minimum  Thrust  of  Rafter§. — Ordinarily,  as  in 
roofs,  the  load  is  not  concentrated,  it  being  that  of  the  fram- 
ing itself.  Here  the  amount  of  the  load  will  be  in  proportion 
to  the  length  of  the  rafter,  and  this  will  increase  with  the 
increase  of  the  angle  of  inclination,  the  span  remaining  the 
same.  So  it  is  seen  that  in  enlarging  the  angle  of  inclina- 
tion to  the  horizon  in  order  to  lessen  the  oblique  thrust,  the 
load  is  increased  in  consequence  of  the  elongation  of  the 
rafter,  thus  increasing  the  oblique  thrust.  Hence  there  is 
an  economical  angle  of  inclination.  A  rafter  will  have  the 
least  oblique  thrust  when  its  angle  of  inclination  .to  the 
horizon  is  35°  16'  nearly.  This  angle  is  attained  very  nearly 
when  the  rafter  rises  8J  inches  per  foot,  or  when  the  height 
B  C  (Fig.  32)  is  to  the  base  A  C  as  8£  is  to  12,  or  as  0-7071  is 
to  i-o. 

77. — Practical  Method  of  Determining  Strain§. — A  com- 
parison of  pressures  in  timbers,  according  to  their  position, 
may  be  readily  made  by  drawing  various  designs  of  framing 
and  estimating  the  several  strains  in  accordance  with  the 
parallelogram  of  forces,  always  drawing  the  triangle  b  d  e 
so  that  the  three  lines  shall  be  parallel  with  the  three  forces 
or  pressures  respectively.  The  length  of  the  lines  forming 
this  triangle  is  unimportant,  but  it  will  be  found  more  con- 
venient if  the  line  drawn  parallel  with  the  known  force  is 
made  to  contain  as  many  inches  as  the  known  force  contains 
pounds,  or  as  many  tenths  of  an  inch  as  pounds,  or  as  many 
inches  as  to  is,  or  tenths  of  an  inch  as  tons;  or,  in  general, 
as  many  divisions  of  any  convenient  scale  as  there  are  units 
of  weight  or  pressure  in  the  known  force.  If  drawn  in  this 
manner,  then  the  number  of  divisions  of  the  same  scale 
found  in  the  other  two  lines  of  the  triangle  will  equal  the 
units  of  pressure  or  weight  of  the  other  two  forces  respect- 


HORIZONTAL  THRUST   OF   RAFTERS.  63 

ively,  and  the  pressures  sought  will  be  ascertained  simply  by 
applying  the  scale  to  the  lines  of  the  triangle. 

For  example,  in  Fig.  23,  the  vertical  line  b  d,  of  the  tri- 
angle, measures  fifty-five  hundredths  of  an  inch  (0-55  inch) ; 
the  line  b  e,  fifty  hundredths  (0-50  inch);  and  the  line  e  d> 
forty  (0-40  inch).  Now,  if  it  be  supposed  that  the  vertical 
pressure,  or  the  weight  suspended  below  b  d,  is  equal  to  55 
pounds,  then  the  pressure  on  A  will  equal  50  pounds,  and 
that  on  B  will  equal  40  pounds  ;  for,  by  the  proportion  above 
stated, 

b  d:  W-.-.b  e:P, 
55  :  55  ::  50:  50; 

and  so  of  the  other  pressure. 

If  a  scale  cannot  be  had  of  equal  proportions  with  the 
forces,  the  arithmetical  process  will  be  shortened  somewhat 
by  making  the  line  of  the  triangle  that  represents  the  known 
weight  equal  to  unity  of  a  decimally  divided  scale,  then  the 
other  lines  will  be  measured  in  tenths  or  hundredths ;  and 
in  the  numerical  statement  of  the  proportions  between  the 
lines  and  forces,  the  first  term  being  unity,  the  fourth  term 
will  be  ascertained  simply  by  multiplying  the  second  and 
third  terms  together. 

For  example,  if  the  three  lines  are  i,  0-7,  and  1-3,  and 
the  known  weight  is  6  tons,  then 

b  d  :    W : :  b  e  :  P   becomes 
I  :  6  : :  0-7  :  P  —  4-2, 

equals  four  and  two  tenths  tons.     Again — 

bd\    W : :  e  d  :  Q  becomes 
I  :6::  1-3:  Q  =  7-8, 

equals  seven  and  eight  tenths  tons. 

78.— Horizontal  Thrust.— In  Fig.  24,  the  weight  ^presses 
the  struts  in  the  direction  of  their  length  ;  their  feet,  n  n, 
therefore,  tend  to  move  in  the  direction  n  o,  and  would  so 
move  were  they  not  opposed  by  a  sufficient  resistance  from 
the  blocks,  A  and  A.  If  a  piece  of  each  block  be  cut  off  at 


64  CONSTRUCTION. 

the  horizontal  line,  a  n,  the  feet  of  the  struts  would  slide 
away  from  each  other  along  that  line,  in  the  direction  na; 
but  if,  instead  of  these,  two  pieces  were  cut  off  at  the  verti- 
cal line,  nb,  then  the  struts  would  descend  vertically.  To 
estimate  the  horizontal  and  the  vertical  pressures  exerted  by 
the  struts,  let  n  o  be  made  equal  (upon  any  scale  of  equal 
parts)  to  the  number  of  tons  with  which  the  strut  is  pressed ; 


FIG.  24. 

construct  the  parallelogram  of  forces  by  drawing  o  c  parallel 
to  an,  and  of  parallel  to  bn;  then  nf  (by  the  same  scale) 
shows  the  number  of  tons  pressure  that  is  exerted  by  the 
strut  in  the  direction  n  a,  and  n  e  shows  the  amount  exerted 
in  the  direction  n  b.  By  constructing  designs  similar  to  this, 
giving  various  and  dissimilar  positions  to  the  struts,  and 
then  estimating  the  pressures,  it  will  be  found  in  every  case 
that  the  horizontal  pressure  of  one  strut  is  exactly  equal  to 
that  of  the  other,  however  much  one  strut  may  be  inclined 
more  than  the  other;  and  also,  that  the  united  vertical 
pressure  of  the  two  struts  is  exactly  equal  to  the  weight  W. 
(In  this  calculation  the  weight  of  the  timbers  has  not  been 
taken  into  consideration,  simply  to  avoid  complication  to 
the  learner.  In  practice  it  is  requisite  to  include  the  weight 
of  the  framing  with  the  load  upon  the  framing.) 

Suppose  that  the  two1  struts,  B  and  B  (Fig.  24),  were 
rafters  of  a  roof,  and  that  instead  of  the  blocks,  A  and  A,  the 
walls  of  a  building  were  the  supports:  then,  to  prevent 


TIES   DESIRABLE   IN   ROOFS.  65 

the  walls  from  being  thrown  over  by  the  thrust  of  B  and  B, 
it  would  be  desirable  to  remove  the  horizontal  pressure. 
Tnis  may  be  done  by  uniting-  the  feet  of  the  rafters  with  a 


u 


FIG.  25. 


rope,  iron  rod,  or  piece  of  timber,  as  in  Fig.  25.  This  figure 
is  similar  to  the  truss  of  a  roof.  The  horizontal  strains  on 
the  tie-beam,  tending  to  pull  it  asunder  in  the  direction  of 
its  length,  may  be  measured  at  the  foot  of  the  rafter,  as  was 
shown  at  Fig*  24 ;  but,  it  can  be  more  readily  and  as  accu- 
rately measured  by  drawing  from  f  and  e  horizontal  lines  to 
the  vertical  line,  b d,  meeting  it  in  o  and  o ;  then/0  will  be 
the  horizontal  thrust  at  B,  and  co  at  A  ;  these  will  be  found 
to  equal  one  another.  When  the  rafters  of  a  roof  are  thus 
connected,  all  tendency  to  thrust  out  the  walls  horizontally 
is  removed,  the  only  pressure  on  them  is  in  a  vertical  direc- 
tion, being  equal  to  the  weight  of  the  roof  and  whatever  it 
has  to  support.  This  pressure  is  beneficial  rather  than 
otherwise,  as  a  roof  having  trusses  thus  formed,  and  the 
trusses  well  braced  to  each  other,  tends  to  steady  the  walls. 

79.— Position  of  Supports.—/^.  26  and  27  exhibit  two 
methods  of  supporting  the  equal  weights,  W  and  W.  Let 
it  be  required  to  measure  and  compare  the  strains  produced 
on  the  pieces,  A  B  and  A  C.  Construct  the  parallelogram  of 
forces,  ebfd,  according  to  Art.  71.  Then  3/will  show  the 


66 


CONSTRUCTION. 


strain  on  A  B,  and  b  e  the  strain  on  A  C.  By  comparing  the 
figures,  bd  being  equal  in  each,  it  will  be  seen  that  the 
strains  in  Fig.  26  are  about  three  times  as  great  as  those  in 


B 


FIG.  27. 


Fig.  27 ;  the  position  of  the  pieces,  A  B  and  A  C,  in  Fig.  27, 
is  therefore  far  preferable. 


FIG.  28. 


80.— The  Composition  of  Force§ :  consists  in  ascertain- 
ing the  direction  and  amount  of  one  force  which  shall  be 
just  capable  of  balancing  tivo  or  more  given  forces,  acting  in 
different  directions.  This  is  only  the  reverse  of  the  resolu- 


STRAINS    INVOLVED    IN   CRANE. 


tion  of  forces ;  and  the  two  are  founded  on  one  and  the 
same  principle,  and  may  be  solved  in  the  same  manner.  For 
example,  let  A  and  B  (Fig.  28)  be  two  pieces  of  timber* 
pressed  in  the  direction  of  their  length  towards  b — A  by  a 
force  equal  to  6  tons  weight,  and  B  9  tons.  To  find  the 
direction  and  amount  of  pressure  they  would  unitedly  exert, 
draw  the  lines  b  e  and  b  f  in  a  line  with  the  axes  of  the 
timbers,  and  make  b  e  equal  to  the  pressure  exerted  by  B, 
viz.,  9 ;  also  make  b  f  equal  to  the  pressure  on  A,  viz.,  6,  and 
complete  the  parallelogram  of  forces  ebfd;  then  bd,  the 
diagonal  of  the  parallelogram,  will  be  the  direction,  and  its 
length,  9-25,  will  be  the  amount,  of  the  united  pressures  of 
A  and  of  B.  The  line  b  d  is  termed  the  resultant  of  the  two 
forces  b  f  and  be.  If  A  and  B  are  to  be  supported  by  one 
post,  Ct  the  best  position  for  that  post  will  be  in  the  direc- 
tion of  the  diagonal  bd\  and  it  will  require  to  be  sufficiently 
strong  to  support  the  united  pressures  of  A  and  of  B,  which 
are  equal  to  9-25  or  9^  tons. 


FIG.  29. 

81.— Another  Example.— Let  Fig.  29  represent  a  piece  of 
Laming  commonly  called  a  crane,  which  is  used  for  hoist- 
ing heavy  weights  by  means  of  the  rope,  B  bf,  which  passes 
over  a  pulley  at  b.  This,  though  similar  to  Figs.  26  and  27,  is, 
however,  still  materially  different.  In  those  figures,  the 
strain  is  in  one  direction  only,  viz.,  from  b  to  d\  but  in  this 
there  are  two  strains,  from  A  to  B  and  from  A  to  W. ^  The 
strain  in  the  direction  A  B  is  evidently  equal  to  that  in  the 


68 


CONSTRUCTION. 


direction  A  W.  To  ascertain  the  best  position  for  the  strut 
A  C,  make  b  e  equal  to  bf,  and  complete  the  parallelogram  of 
forces  ebfd;  then  draw  the  diagonal  bd,  and  it  will  be  the 
position  required.  Should  the  foot,  C,  of  the  strut  be  placed 
either  higher  or  lower,  the  strain  on  A  C  would  be  in- 
creased. In  constructing  cranes,  it  is  advisable,  in  order 
that  the  piece  B  A  may  be  under  a  gentle  pressure,  to  place 
the  foot  of  the.  strut  a  trifle  lower  than  where  the  diagonal 
bd  would  indicate,  but  never  higher. 


W 


FIG.  30. 

82.— Tie§  and  Strut§.— Timbers  in  a  state  of  tension  are 
called  ties,  while  such  as  are  in  a  state  of  compression  are 
termed  struts.  This  subject  can  be  illustrated  in  the  follow- 
ing manner : 

Let  A  and  B  (Fig.  30)  represent  beams  of  timber  sup- 
porting the  weights  W,  W,  and  W ' ;  A  having  but  one  sup- 
port, which  is  in  the  middle  of  its  length,  and  B  two,  one  at 
each  end.  To  show  the  nature  of  the  strains,  let  each  beam 
be  sawed  in  the  middle  from  a  to  b.  The  effects  are  obvious: 
the  cut  in  the  beam  A  will  open,  whereas  that  in  B  will 
close.  If  the  weights  are  heavy  enough,  the  beam  A  will 
break  at  b ;  while  the  cut  in  B  will  be  closed  perfectly  tight 
at  a,  and  the  beam  be  very  little  injured  by  it.  But  if,  on 
the  other  hand,  the  cuts  be  made  in  the  bottom  edge  of  the 
timbers,  from  c  to  b,  B  will  be  seriously  injured,  while  A 
will  scarcely  be  affected.  .  By  this  it  appears  evident  that, 
in  a  piece  of  timber  subject  to  a  pressure  across  the  direction 
of  its  length,  the  fibres  are  exposed  to  contrary  strains.  If 
the  timber  is  supported  at  both  ends,  as  at  B,  those  from  the 
top  edge  down  to  the  middle  are  compressed  in  the  direction 


TIES    AND    STRUTS.  69 

of  their  length,  while  those  from  the  middle  to  the  bottom 
edge  are  in  a  state  of  tension  ;  but  if  the  beam  is  supported 
as  at  A,  the  contrary  effect  is  produced  ;  while  the  fibres  at 
the  middle  of  either  beam  are  not  at  all  strained.  The  strains 
in  a  framed  truss  are  of  the  same  nature  as  those  in  a  single 
beam.  The  truss  for  a  roof,  being  supported  at  each  end, 
has  its  tie-beam  in  a  state  of  tension,  while  its  rafters  are 
compressed  in  the  direction  of  their  length.  By  this,  it 
appears  highly  important  that  pieces  in  a  state  of  tension 
should  be  distinguished  from  such  as  are  compressed,  in 
order  that  the  former  may  be  preserved  continuous.  A  strut 
may  be  constructed  of  two  or  more  pieces  ;  yet,  where  there 
are  many  joints,  it  will  not  resist  compression  so  well. 

83.— To  Distinguish  Ties  from  Struts.— This  may  be  done 
by  the  following  rule.  In  Fig.  22-B,  the  timbers  C  and  D  are 
the  sustaining  forces,  and  the  weight  £Fis  the  straining  force ; 
and  if  the  support  be  removed,  the  straining  force  would 
move  from  the  point  of  support  b  towards  d.  Let  it  be 
required  to  ascertain  whether  the  sustaining  forces  arc 
stretched  or  pressed  by  the  straining  force.  Rule :  Upon  the 
direction  of  the  straining  force  b  d,  as  a  diagonal,  construct 
a  parallelogram  ebfd  whose  sides  shall  be  parallel  with  the 
direction  of  the  sustaining  forces  C  and  D\  through  the 
point  b  draw  a  line  parallel  to  the  diagonal  ef\  this  may 
then  be  called  the  dividing  line  between  ties  and  struts. 
Because  all  those  supports  which  are  on  that  side  of  the 
dividing  line  which  the  straining  force  would  occupy  if 
unresisted  are  compressed,  while  those  on  the  other  side  of 
the  dividing  line  are  stretched. 

In  Fig.  22-B,  the  supports  are  both  compressed,  being  on 
that  side  of  the  dividing  line  which  the  straining  force  would 
occupy  if  unresisted.  In  Figs.  26  and  27,  in  which  A  B  and 
A  C  are  the  sustaining  forces,  A  C  is  compressed,  whereas 
A  B  is  in  a  state  of  tension ;  A  C  being  on  that  side  of  the 
line  h  i  which  the  straining  force  would  occupy  if  unresisted, 
and  A  B  on  the  opposite  side.  The  place  of  the  latter  might 
be  supplied  by  a  chain  or  rope.  In  Fig.  25,  the  foot  of  the 
rafter  at  A  is  sustained  by  two  forces,  the  wall  and  the  tie- 


70  CONSTRUCTION. 

beam,  one  perpendicular  and  the  other  horizontal:  the 
direction  of  the  straining  force  is  indicated  by  the  line  b  a. 
The  dividing  line  h  i,  ascertained  by  the  rule,  shows  that  the 
wall  is  pressed  and  the  tie-beam  stretched. 


FIG.  31. 


84.— Another  Example.— Let  E  A  B  F  (Fig.  31)  represent 
a  gate,  supported  by  hinges  at  A  and  E.  In  this  case,  the 
straining  force  is  the  weight  of  the  materials,  and  the  direc- 
tion of  course  vertical.  Ascertain  the  dividing  line  at  the 
several  points,  G,  B,  /,  J,  H,  and  F.  It  will  then  appear  that 
the  force  at  G  is  sustained  by  A  G  and  G  E,  and  the  dividing 
line  shows  that  th'e  former  is  stretched  and  the  latter  com- 
pressed. The  force  at  H  is  supported  by  A  H  and  HE — the 
former  stretched  and  the  latter  compressed.  The  force  at  B 
is  opposed  by  H  B  and  A  B,  one  pressed,  the  other  stretched. 
The  force  at  F  is  sustained  by  G  F  and  FE,  G  F  being 
stretched  and  F  E  pressed.  By  this  it  appears  that  A  B  is  in 
a  state  of  tension,  and  E  F  of  compression  ;  also,  that  A  H 
and  G  F  are  stretched,  while  B  H  and  G  E  are  compressed  : 
which  shows  the  necessity  of  having  A  H  and  G  F  each  in 
one  whole  length,  while  B  H  and  G  E  may  be,  as  they  are 
shown,  each  in  two  pieces.  The  force  at  J  is  sustained  by 
GJ  and  J H,  the  former  stretched  and  the  latter  compressed. 
The  piece  C  D  is  neither  stretched  nor  pressed,  and  could 
be  dispensed  with  if  the  joinings  at  J  and  /  could  be  made 


TO    FIND    THE   CENTRE   OF   GRAVITY.  71 

as  effectually  without  it.  In  case  A  B  should  fail,  then  C  D 
would  be  in  a  state  of  tension. 

85.  —  Centre  of  Gravity.  —  The  centre  of  gravity  of  a  uni- 
form prism  or  cylinder  is  in  its  axis,  at  the  middle  of  its 
length  ;  that  of  a  triangle  is  in  a  line  drawn  from  one  angle 
to  the  middle  of  the  opposite  side,  and  at  one  third  of  the 
length  of  the  line  from  that  side  ;  that  of  a  right-angled  tri- 
angle, at  a  point  distant  from  the  perpendicular  equal  to  one 
third  of  the  base,  and  distant  from  the  base  equal  to  one 
third  of  the  perpendicular  ;  that  of  a  pyramid  or  cone,  in 
the  axis  and  at  one  quarter  of  the  height  from  the  base. 

The  centre  of  gravity  of  a  trapezoid  (a  four-sided,  figure 
having  only  two  of  its  sides  parallel)  is  in  a  line  joining  the 
centres  of  the  two  parallel  sides,  and  at  a  distance  from  the 
longest  of  the  parallel  sides  equal  to  the  product  of  the 
length  in  the  sum  of  twice  the  shorter  added  to  the  longer 
of  the  parallel  sides,  divided  by  three  times  the  sum  of  the 
two  parallel  sides.  Algebraically  thus  : 

,_'_(2_ 
-  3  («  + 

where  d  equals  the  distance  from  the  longest  of  the  parallel 
sides,  /  the  length  of  the  line  joining  the  two  parallel  sides, 
and  a  the  shorter  and  b  the  longer  of  the  parallel  sides. 

Example.  —  A  rafter  25  feet  long  has  the  larger  end  14 
inches  wide,  and  the  smaller  end  10  inches  wide:  how  far 
from  the  larger  end  is  the  centre  of  gravity  located  ? 

Here  /  =  25,  a  —  |f,  and  b  =  |f, 


-       _  .  25_x_       =  25x34, 

"   W±P)   '        3(W  +  «)        "   3xff       3x24  - 

—  °  =11-8  =  11  feet  9!  inches  nearly. 

In  irregular  bodies  with  plain  sides,  the  centre  of  gravity 
may  be  found  by  balancing  them  upon  the  edge  of  a  prism 

—  upon  the  edge  of  a  table  —  in  two  positions,  making  a  line 
each  time  upon  the  body  in  a  line  with  the  edge  of  the  prism, 
and  the  intersection  of  those  lines  will  indicate  the  point  re- 


72  CONSTRUCTION. 

quired.  Or  suspend  the  article  by  a  cord  or  thread  attached 
to  one  corner  or  edge  ;  also  from  the  same  point  of  suspen- 
sion hang  a  plumb-line,  and  mark  its  position  on  the  face  of 
the  article;  again,  suspend  the  article  from  another  corner 
or  side  (nearly  at  right  angles  to  its  former  position),  and 
mark  the  position  of  the  plumb-line  upon  its  face ;  then  the 
intersection  of  the  two  lines  will  be  the  centre  of  gravity. 


FIG.  32. 

86. — Effect  of  the  Weight  of  Inclined  Beam§. — An  in- 
clined post  or  strut  supporting  some  heavy  pressure  applied 
at  its  upper  end,  as  at  Fig.  25,  exerts  a  pressure  at  its  foot  in 
the  direction  of  its  length,  or  nearly  so.  But  when  such  a 
beam  is  loaded  uniformly  over  its  whole  length,  as  the  rafter 
of  a  roof,  the  pressure  at  its  foot  varies  considerably  from 
the  direction  of  its  length.  For  example,  let  A  B  (Fig.  32) 
be  a  beam  leaning  against  the  wall  B  c,  and  supported  at  its 
foot  by  the  abutment  A,  in  the  beam  A  c,  and  let  o  be  the 
centre  of  gravity  of  the  beam.  Through  o  draw  the  verti- 
cal line  b  dy  and  from  B  draw  the  horizontal  line  B  b,  cutting 
b  dm  b\  join  b  and  A,  and  b  A  will  be  the  direction  of  the 
thrust.  To  prevent  the  beam  from  loosing  its  footing,  the 
joint  at  A  should  be  made  at  right  angles  to  b  A.  The 
amount  of  pressure  will  be  found  thus :  Let  b  d  (by  any  scale 
of  equal  parts)  equal  the  number  of  tons  upon  the  beam 
A  B\  draw  d  e  parallel  to  B  b  ;  then  /;  c  (by  the  same  scale) 
equals  the  pressure  in  the  direction  b  A  ;  and  e  d  the  pres- 
sure against  the  wall  at  B— and  also  the  horizontal  thrust  at 
A^  as  these  are  always  equal  in  a  construction  of  this  kind. 

The  horizontal  thrust  of  an  inclined  beam  (Fig.  32)— the 
effect  of  its  own  weight— may  be  calculated  thus : 

Rule.— Multiply   the  weight  of  the  beam  in  pounds  by 


THRUST    OF   INCLINED   BEAMS.  73 

its  base,  A  C,  in  feet,  and  by  the  distance  in  feet  of  its  centre 
of  gravity,  o  (see  Art.  85),  from  the  lower  end,  at  A,  and 
divide  this  product  by  the  product  of  the  length,  A  B,  into 
the  height,  B  C,  and  the  quotient  will  be  the  horizontal 

thrust  in   pounds.      This    may  be   stated   thus :  H  = — , 

where  d  equals  the  distance  of  the  centre  of  gravity,  0,  from 
the  lower  end  ;  b  equals  the  base,  A  C ;  iv  equals  the  weight 
of  the  beam  ;  h  equals  the  height,  D  C ;  /equals  the  length 
of  the  beam  ;  and  H  equals  the  horizontal  thrust. 

Example. — A  beam  20  feet  long  weighs  300  pounds;  its 
centre  of  gravity  is  at  9  feet  from  its  lower  end  ;  it  is  so  in- 
clined that  its  base  is  16  feet  and  its  height  12  feet :  what  is 
the  horizontal  thrust  ? 

TT        d  b  w ,  o  x  16  x  300       Q  x  4  x  25 

Here  —  — —  becomes  — —  —  - — —  0x4x5 

hi  12x20  5 

=  180  =H  =.  the  horizontal  thrust. 

This  rule  is  for  cases  where  the  centre  of  gravity  does 
not  occur  at  the  middle  of  the  length  of  the  beam,  although 
it  is  applicable  when  it  does  occur  at  the  middle ;  yet  a 
shorter  rule  will  suffice  in  this  case,  and  it  is  thus: 

Rule. — Multiply  the  weight  of  the  rafter  in  pounds  by 
the  base,  A  C  (Fig.  32),  in  feet,  and  divide  the  product  by 
twice  the  height,  B  C,  in  feet,  and  the  quotient  will  be  the 
horizontal  thrust,  when  the  centre  of  gravity  occurs  at  the 
middle  of  the  beam. 

If  the  inclined  beam  is  loaded  with  an  equally  distributed 
load,  add  this  load  to  the  weight  of  the  beam,  and  use  this 
total  weight  in  the  rule  instead  of  the  weight  of  the  beam. 
And  generally,  if  the  centre  of  gravity  of  the  combined 
weights  of  the  beam  and  load  does  not  occur  at  the  centre 
of  the  length  of  the  beam,  then  the  former  rule  is  to  be  used. 

In  Fig.  33,  two  equal  beams  are  supported  at  their  feet  by 
the  abutments  in  the  tie-beam.  This  case  is  similar  to  the 
last ;  for  it  is  obvious  that  each  beam  is  in  precisely  the 
position  of  the  beam  in  Fig.  32.  The  horizontal  pressures  at 
B,  being  equal  and  opposite,  balance  one  another ;  and  their 
horizontal  thrusts  at  the  tie-beam  are  also  equal.  (See  Art. 


CONSTRUCTION. 


78— Fig.  25.)  When  the  height  of  a  roof  (Fig.  33)  is  one 
fourth  of  the  span,  or  of  a  shed  (Fig.  32)  is  one  halt  the 
span,  the  horizontal  thrust  of  a  rafter,  whose  centre  of  grav- 


FIG.  33. 

ity  is  at  the  middle  of  its  length,  is  exactly  equal  to  the 
weight  distributed  uniformly  over  its  surface. 

In  shed  or  lean-to  roofs,  as  Fig.  32,  the  horizontal  pressure 
will  be  entirely  removed  if  the  bearings  of  the  rafters,  as  A 
and  B  (Fig.  34),  are  made  horizontal — provided,  however, 


FIG.  34. 

that  the  rafters  and  other  framing  do  not  bend  between  the 
points  of  support.  If  a  beam  or  rafter  have  a  natural  curve, 
the  convex  or  rounding  edge  should  be  laid  uppermost. 

87. — Effect  of  Load  011  Beam. — The  strain  in  a  uniformly 
loaded  beam,  supported  at  each  end,  is  greatest  at  the 
middle  of  its  length.  Hence  mortices,  large  knots,  and  other 
defects  should  be  kept  as  far  as  possible  from  that  point ; 
and  in  resting  a  load  upon  a  beam,  as  a  partition  upon  a 
floor-beam,  the  weight  should  be  so  adjusted,  if  possible, 
that  it  will  bear  at  or  near  the  ends. 

Twice  the  weight  that  will  break  a  beam,  acting  at  the 
centre  of  its  length,  is  required  to  break  it  when  equally 


VARYING   PRESSURE   ON   BEARINGS.  75 

distributed  over  its  length  ;  and  precisely  the  same  deflec- 
tion or  .rag- 'will  be  produced  on  a  beam  by  a  load  equally 
distributed  that  five  eighths  ot  the  load  will  produce  if  act- 
ing at  the  centre  of  its  length. 

88. — Effect  011  Bern-ings. — When  a  uniformly  loaded 
beam  is  supported  at  each  end  on  level  bearings  (the  beam 
itself  being  either  horizontal  or  inclined),  the  amount  of 
pressure  caused  by  the  load  on  each  point  of  support  is 
equal  to  one  half  the  load  ;  and  this  is  also  the  ase  when 
the  load  is  concentrated  at  the  middle  of  the  beam,  or  has 
its  centre  of  gravity  at  the  middle  of  the  beam ;  but  when 
the  load  is  unequally  distributed,  or  concentrated  so  that  its 
centre  of  gravity  occurs  at  some  other  point  than  the  middle 
of  the  beam,  then  the  amount  of  pressure  caused  by  the 
load  on  one  of  the  points  of  support  is  unequal  to  that  on 
the  other.  The  precise  amount  on  each  may  be  ascertained 
by  the  following  rule. 

Rule.—  Multiply  the  weight   W  (Fig.  35)  by  its  distance, 
C B,  from  its  nearest  point  of  support,  By  and  divide  the  pro-, 
duct  by  the  length,  A  B,  of  the  beam,  and  the  quotient  will 


FIG  35. 

be  the  amount  of  pressure  on  the  remote  point  of  support,  A. 
Again,  deduct  this  amount  from  the  weight  W,  and  the  re- 
mainder will  be  the  amount  of  pressure  on  the  near  point  of 
support,  B ;  or,  multiply  the  weight  W  by  its  distance,  A  C, 
from  the  remote  point  of  support,  A,  and  divide  the  pro- 
duct by  the  length,  A  B,  and  the  quotient  will  be  the  amount 
of  pressure  on  the  near  point  of  support,  B. 

When  /  equals  the  length  between  the  bearings  A  and  B, 
n  —  AC,  m  —  C  B,  and  W  —  the  load  ;  then 


7g  CONSTRUCTION. 

v/  I 

JL_/-  —  A  =  the  amount  of  pressure  at  A,  and 

.    ?  =  ,5  =  the  amount  of  pressure  at  ^?. 

Example.— A  beam  20  feet  long  between  the  bearings 
has  a  load  of  100  pounds  concentrated  at  3  feet  from  one  of 
the  bearings :  what  is  the  portion  of  this  weight  sustained  by 
each  bearing  ? 

Here  W—  100 ;  «,  17 ;  m,  3 ;  and  /,  20. 


W  m 
Hence  A=-j- 


Load  on  A  =  15  pounds. 
Load  on  ^  =  85  pounds, 
Total  weight  =  100  pounds. 

RESISTANCE    OF    MATERIALS. 

89.  —  Weight—  Strength.  —  Preliminary  to  designing  a  roof- 
truss  or  other  piece  of  framing,  a  knowledge  of  two  subjects 
is  essential  :  one  is,  the  effect  of  gravity  acting  upon  the 
various  parts  of  the  intended  structure  ;  the  other,  the  power 
of  resistance  possessed  by  the  materials  of  which  the  framing 
is  to  be   constructed.      The   former  subject   having  been 
treated  of  in  the  preceding  pages,  it  remains  now  to  call  at- 
tention to  the  latter. 

90.  —  Quality  of  Materials.—  Materials  used  in  construc- 
tion  are   constituted    in    their   structure    either   of    fibres 
(threads)  or  of  grains,  and  are  termed,  the  former  fibrous, 
the  latter  granular.     All  woods  and  wrought   metals   are 
fibrous,  while  cast  iron,  stone,  glass,  etc.,  are  granular.    The 
strength  of  a  granular  material  lies  in  the  power  of  attrac- 
tion acting  among  the  grains  of  matter  of  which  the  mate- 
rial is  composed,  by  which  it  resists  any  attempt  to  separate 
its  grains  or  particles  of  matter.     A  fibre  of  wood  or  ot 


UNIVERSITY 


THE   THREE   KINDS   OF   ST 


wrought  metal  has  a  strength  by  which  it  resists  being  com- 
pressed or  shortened,  and  finally  crushed  ;  also  a  strength 
by  which  it  resists  being  extended  or  made  longer,  and 
finally  sundered.  There  is  another  kind  of  strength  in  a 
fibrous  material  :  it  is  the  adhesion  of  one  fibre  to  another 
along  their  sides,  or  the  lateral  adhesion  of  the  fibres. 

91.  —  Manner  of  Rc§isting.  —  In  the  strain  applied  to  a 
post  supporting  a  weight  imposed  upon  it  (Fig.  36),  we 
have  an  instance  of  an  essay  to  shorten  the  fibres  of  which 
the  timber  is  composed.  The  strength  of  the  timber  in 
this  case  is  termed  the  resistance  to  compression.  In  the  strain 
on  a  piece  of  timber  like  a  king-post  or  suspending  piece 
(A,  Fig.  37),  we  have  an  instance  of  an  essay  to  extend  or 
lengthen  the  fibres  of  the  material.  The  strength  here  ex- 
hibited is  termed  the  resistance  to  tension.  When  a  piece  of 
timber  is  strained  like  a  floor-beam  or  any  horizontal  piece 


FIG  37. 


FIG  38. 


carrying  a  load  (Fig.  38).  we  have  an  instance  in  which  the 
two  strains  of  compression  and  tension  are  both  brought 
into  action  ;  the  fibres  of  the  upper  portion  of  the  beam  be- 
ing compressed,  and  those  of  the  under  part  being  stretched. 


78  CONSTRUCTION. 

This  kind  of  strength  of  timber  is  termed  resistance  to  cross- 
strains.  In  each  of  these  three  kinds  of  strain  to  which  tim- 
ber is  subjected,  the  power  of  resistance  is  in  a  measure  due 
to  the  lateral  adhesion  of  the  fibres,  not  so  much  perhaps  in 
the  simple  tensile  strain,  yet  to  a  considerable  degree  in  the 
compressive  and  cross  strains.  But  the  power  of  timber, 
by  which  it  resists -a  pressure  acting  compressively  in  the 
direction  of  the  length  of  the  fibres,  tending  to  separate  the 
timber  by  splitting  off  a  part,  as  in  the  case  of  the  end  of  a 
tie-beam,  against  which  the  foot  of  the  rafter  presses,  is 
wholly  due  to  the  lateral  adhesion  of  the  fibres. 

92. — Strength  and  Stiffne§§. — The  strengtJi  of  materials 
is  their  power  to  resist  fracture,  while  the  stiffness  of  mate- 
rials is  their  capability  to  resist  deflection  or  sagging.  A 
knowledge  of  their  strengtJi  is  useful,  in  order  to  determine 
their  limits  of  size  to  sustain  given  weights  safely  ;  but  a 
knowledge  of  their  stiffness  is  more  important,  as  in  almost 
all  constructions  it  is  desirable  not  only  that  the  load  be  safely 
sustained,  but  that  no  appearance  of  weakness  be  manifested 
by  any  sensible  deflection  or  sagging. 

93. — Experiments  :  Constants —  In  the  investigation  of 
the  laws  applicable  to  the  resistance  of  materials,  it  is  found 
that  the  dimensions — length,  breadth,  and  thickness — bear 
certain  relations  to  the  weight  or  pressure  to  which  the 
piece  is  subjected.  These  relations  are  general ;  they  exist 
quite  independently  of  the  peculiarities  of  any  specific  piece 
of  material.  These  proportions  between  the  dimensions 
and  the  load  are  found  to  exist  alike  in  wood,  metal,  stone, 
and  glass,  or  other  material.  One  law  applies  alike  to  all 
materials  ;  but  the  capability  of  materials  to  resist  differs  in 
accordance  with  the  compactness  and  cohesion  of  particles, 
and  the  tenacity  and  adhesion  of  fibres,  those  qualities  upon 
which  depends  the  superiority  of  one  kind  of  material  over 
another.  The  capability  of  each  particular  kind  of  material 
is  ascertained  by  experiments,  made  upon  several  specimens, 
and  an  average  of  the  results  thus  obtained  is  taken  as  an 
index  of  the  capability  of  that  material,  and  is  introduced 
in  the  rules  as  a  constant  number,  each  specific  kind  of  ma- 


VALUES    OF   WOODS    FOR   COMPRESSION. 


79 


terial  having-  its  own  special  constant,  obtained  by  ex- 
perimenting- on  specimens  of  that  peculiar  material.  The 
results  of  experiments  made  to  test  the  resistance  of  various 
materials  useful  in  construction — their  capability  to  resist 
the  three  strains  before  named — will  now  be  introduced. 

94. — Resistance  to  Compression. — The  following  table 
exhibits  the  results  of  experiments  made  to  test  the  resist- 
ance to  compression  of  such  woods  as  are  in  common  use  in 
this  country  for  the  purposes  of  construction. 

TABLE  I. — RESISTANCE  TO  COMPRESSION. 


MATERIAL. 

>, 

"> 
Z 
O 
u 

te 

1 

To  crush  fibres  ~ 
longitudinally.  • 

To  separate  fibres 

by  sliding.  fS 

To  crush  fibres  trans- 
versely 3^  inch  deep. 

Value  of  .Pin  Rules. 
Sensible  Impres-  ^ 
sion. 

Georgia  Pine  

0-611 

Pounds 
per  inch. 
95OO 

Pounds 
per  inch. 
840 

Pounds 
per  inch. 
225O 

QOO 

0-762 

II7OO 

1160 

28OO 

1  1  2O 

White  Oak  

O-  774 

8000 

I25O 

2650 

IO6O 

Spruce  

0-360 

7850 

540 

650 

26O 

White  Pine 

O-^88 

6650 

480 

800 

32O 

Hemlock       

O-423 

57OO 

370 

800 

32O 

White  Wood  

o-  307 

3400 

800 

320 

Chestnut 

O-4QI 

6700 

1250 

5OO 

Ash    

o-  517 

5850 

3'OO 

1240 

Maple  

O-574 

8450 

2700 

I080 

Hickory 

O-877 

13750 

4100 

1640 

Cherry 

O*4Q4 

QO5O 

2500 

IOOO 

Black  Walnut  

O-42I 

7800 

2IOO 

840 

Mahogany  (St.  Domingo)  
(Bay  Wood)  

0-837 
O-43Q 

Il6oO 
4900 

• 

5700 
I7OO 

2280 
680 

Live  Oak 

o  •  016 

IIIOO 

6800 

2720 

Lignum  Vita?                           .  . 

\J     yj.w 

1-282 

I2IOO 

7700 

3080 

The  resistance  of  timber  of  the  same  name  varies  much ; 
depending  as  it  obviously  must  on  the  soil  in  which  it  grew 
on  its  age  before  and  after  cutting,  on  the  time  of  year 
when  cut,  and  on  the  manner  in  which  it  has  been  kept  since 
it  was  cut.  And  of  wood  from  the  same  tree  much  depends 
upon  its  location,  whether  at  the  butt  or  towards  the  limbs, 
and  whether  at  the  heart  or  of  the  sap,  or  at  a  point  mid- 
way from  the  centre  to  the  circumference  of  the  tree.  The 


8O  CONSTRUCTION. 

pieces  submitted  to  experiment  were  of  ordinary  good 
quality,  such  as  would  be  deemed  proper  to  be  used  in 
framing.  The  prisms  crushed  were  generally  small,  about 
2  inches  long,  and  from  I  inch  to  i-J  inches  square ;  some 
were  wider  one  way  than  the  other,  but  all  containing  in 
area  of  cross  section  from  I  to  2  inches.  The  weight  given 
in  the  table  is  the  average  weight  per  superficial  inch. 

Of  the  first  six  woods  named,  there  were  nine  specimens 
of  each  tested ;  of  the  others,  generally  three  specimens. 

The  results  for  the  first  six  woods  named  are  taken 
from  the  author's  work  on  Transverse  Strains,  published  by 
John  Wiley  &  Sons,  New  York.  The  results  for  these  six 
woods,  as  well  as  those  for  all  the  others  named  in  the  table, 
were  obtained  by  experiments  carefully  made  by  the  author. 
The  first  six  woods  named  were  tested  in  1874  and  1876,  and 
upon  a  testing  machine,  in  which  the  power  is  transmitted 
to  the  pieces  tested,  by  levers  acting  upon  knife-edges. 
For  a  description  of  this  machine,  see  Transverse  Strains, 
Art.  704.  The  woods  named  in  the  table,  other  than  the 
first  six,  were  tested  some  twenty  years  since,  and  upon  a 
hydraulic  press,  which,  owing  to  friction,  gave  results  too 
low. 

The  results,  as  thus  ascertained,  were  given  to  the  public 
in  the  7th  edition  of  this  work,  in  1857.  In  the  present  edi- 
tion, the  figures  in  Table  I.,  for  these  woods,  are  those 
which  have  resulted  by  adding  to  the  results  given  by  the 
hydraulic  press  a  certain  quantity  thought  to  be  requisite 
to  compensate  for  the  loss  by  friction.  Thus  corrected,  the 
figures  in  the  table  may  be  taken  as  sufficiently  near  approx- 
imations for  use  in  the  rules, — although  not  so  trustworthy 
as  the  results  given  for  the  first  six  woods  named,  as  these 
were  obtained  upon  a  superior  testing  machine,  as  above 
stated. 

In  the  preceding  table,  the  second  column  contains  the 
specific  gravity  of  the  several  kinds  of  wood,  showing  their 
comparative  density.  The  weight  in  pounds  of  a  cubic  foot 
of  any  kind  of  wood  or  other  material  is  equal  to  its 
specific  gravity  multiplied  by  62-5,  this  number  being  the 
weight  in  pounds  of  a  cubic  foot  of  water.  The  third  column 


EXPLANATION    OF   TABLE   I.  8 1 

contains  the  weight  in  pounds  required  to  crush  a  prism 
having  a  base  of  one  inch  square ;  the  pressure  applied  to 
the  fibres  longitudinally.  In  practice,  it  is  usual  never  to 
load  material  exposed  to  compression  with  more  than  one 
fourth  of  the  crushing  weight,  and  generally  with  from  one 
sixth  to  one  tenth  only.  The  fourth  column  contains  the 
weight  in  pounds  wrhich,  applied  in  line  with  the  length  of  the 
fibres,  is  required  to  force  off  a  part  of  the  piece,  causing  the 
fibres  to  separate  by  sliding,  the  surface  separated  being  one 
inch  square.  The  fifth  column  contains  the  weight  in  pounds 
required  to  crush  the  piece  when  the  pressure  is  applied  to  the 
fibres  transversely,  the  piece  being  one  inch  thick,  and  the 
surface  crushed  being  one  inch  square,  and  depressed  one 
twentieth  of  an  inch  deep.  The  sixth  column  contains  the 
value  of  P  in  the  rules ;  P  being  the  weight  in  pounds,  ap- 
plied to  the  fibres  transversely,  which  is  required  to  make  a 
sensible  impression  one  inch  square  on  the  side  of  the  piece, 
this  being  the  greatest  weight  that  would  be  proper  for  a 
post  to  be  loaded  with  per  inch  surface  of  bearing,  resting 
on  the  side  of  the  kind  of  wood  set  opposite  in  the  table.  A 
greater  weight  would,  in  proportion  to  the  excess,  crush  the 
side  of  the  wood  under  the  post,  and  proportionably  derange 
the  framing,  if  not  cause  a  total  failure.  It  will  be  observed 
that  the  measure  of  this  resistance  is  useful  in  limiting  the 
load  on  a  post  according  to  the  kind  of  material  contained, 
not  in  the  post,  but  in  the  timber  upon  which  the  post  presses. 

95. — He§i§tance  to  Tension. — The  resistance  of  materials 
to  the  force  of  stretching,  as  exemplified  in  the  case  of  a 
rope  from  which  a  weight  is  suspended,  is  termed  the  resist- 
ance to  tension.  In  fibrous  materials,  this  force  will  be  differ- 
ent in  the  same  specimen,  in  accordance  \vith  the  direction 
•  in  which  the  force  acts,  whether  in  the  direction  of  the  length 
of  the  fibres  or  at  right  angles  to  the  direction  of  their  length. 
It  has  been  found  that,  in  hard  woods,  the  resistance  in  the 
former  direction  is  about  eight  to  ten  times  what  it  is  in  the 
latter;  and  in  soft  woods,  straight,  grained,  such  as  white 
pine,  the  resistance  is  from  sixteen  to  twenty  times.  A 
knowledge  of  the  resistance  in  the  direction  of  the  1 
the  most  useful  in  practice. 


82 


CONSTRUCTION. 


In  the  following  table  are  recorded  the  results  of  ex- 
periments made  to  test  this  resistance  in  some  of  the  woods 
in  common  use,  and  also  in  iron,  cast  and  wrought.  Each 
specimen  of  the  woods  was  turned  cylindrical,  and  about  2 
inches  diameter,  and  then  the  middle  part  reduced  to  about 
f  of  an  inch  diameter,  at  the  middle  of  the  reduced  part, 
and  thence  gradually  increased  toward  each  end,  where  it 
was  considerably  larger  at  its  junction  with  the  enlarged 
end.  The  results,  in  the  case  of  the  iron  and  of  the  first  six 
woods  named,  are  taken  from  the  author's  work,  Trans- 
verse Strains,  Table  XX.  Experiments  were  made  upon 
the  other  three  woods  named  by  a  hydraulic  press,  some 
twenty  years  since,  and  the  results  were  first  published  in 
the  7th  edition  of  this  work,  in  1857.  These  results,  owing 
to  friction,  were  too  low.  Adding  to  them  what  is  supposed 
to  be  the  loss  by  friction  of  the  machine,  the  results  thus 
corrected  are  what  are  given  for  these  three  woods  in  the 
following  table,  and  may  be  taken  as  fair  approximations, 
but  are  not  so  trustworthy  as  the  figures  given  for  the  other 
six  woods  and  for  the  metals. 


TABLE  II. — RESISTANCE  TO  TENSION. 


MATERIAL. 

Specific 
Gravity. 

T. 

Pounds      re- 
quired to  rup- 
ture one  inch 
square. 

Georgia  Pine  

O-6^ 

16000 

Locust  

O'7Q4 

24800 

White  Oak  

O  •  77J. 

IQCQO 

O  4^2 

TOCOO 

White  Pine  

Q.  4^8 

I2OCO 

Hemlock    f  

o  •  402 

87OO 

Hickory  

O-  7^1 

26OCO 

Maple  

O-  6o4 

2OOOO 

Ash  

O-6o8 

I5OOO 

Cast  Iron,  American  )  from 

6  •  Q44 

English      f  to 

7  •  c84 

1  7OOO 

Wrought  Iron,  American  )  from 

7  •  6OO 

English      y  to 

7  •  7Q2 

5OOOO 

The  figures  in  the  table  denote  the  ultimate  capability  of 
a  bar  one  inch  square,  or  the  weight  in  pounds  required  to 


VALUES   OF   MATERIALS  FOR  CROSS-STRAINS. 


produce  rupture.  Just  what  portion  of  this  should  be  taken 
as  the  safe  capability  will  depend  upon  the  nature  of  the 
strain  to  which  the  material  is  to  be  exposed.  In  practice  it 
is  found  that,  through  defects  in  workmanship,  the  attach- 
ments may  be  so  made  as  to  cause  the  strain  to  act  along  one 
side  of  the  piece,  instead  of  through  its  axis;  and  that  in  this 
case  fracture  will  be  produced  with  one  third  of  the  strain 
that  can  be  sustained  through  the  axis.  Due  to  this  and 
other  contingencies,  it  is  usual  to  subject  materials  exposed 
to  tensile  strain  with  only  from  one  sixth  to  one  tenth  of 
the  breaking  weight. 

96. — Resistance  to  Transverse  Strains. — In  the  follow- 
ing table  are  recorded  the  results  of  experiments  made  to 
test  the  capability  of  the  various  materials  named  to  resist 
the  effects  of  transverse  strain.  The  figures  are  taken  from 
the  author's  work,  Transverse  Strains,  before  referred  to. 

TABLE  III. — TRANSVERSE  STRAINS. 


MATERIAL. 

* 

Resistance 
to 
Rupture. 

F. 

Resistance 
to 
Flexure. 

e. 

Extension 
of 
Fibres. 

a, 

Margin 
for 
Safety. 

Georgia.  Pine     

850 

5QOO 

•00109 

I-84 

I2OO 

5050 

•OOI5 

2-20 

White  Oak                          

6"iO 

3IOO 

•00086 

3'39 

55° 

35OO 

•  00098 

2-23 

White  Pine  

500 

2900 

•0014 

I-7I 

Hemlock                 

450 

2800 

•00095 

2-35 

White  Wood      

600 

3450 

•00096 

2-52 

480 

2550 

•00103 

2-54 

A«*h                             

QOO 

4OOO 

•OOIII 

2    82 

Maple               

IIOO 

5I50 

•  0014 

2-12 

1050 

3850 

•0013 

2-QI 

Cherry                       

650 

2850 

•001563 

2-03 

Black  Walnut      

750 

3900 

•00104 

2-57 

Mahogany  (St   Domingo)      .... 

650 

3600 

00116 

2-16 

(Bay  Wood)    

850 

475° 

•00109 

2-28 

Cast  Iron,  American  .  .    
"          English             

2500 

2IOO 

50000 
40000 

26OO 

62000 

"           English 

iqoo 

60000 

Steel    in  Bars       •        

6000 

70000 

1 

200 

59 

33 

'  '        pressed                     

37 

Marble   East  Chester 

147 

84  CONSTRUCTION. 

The  figures  in  the  second  column,  headed  B,  denote  the 
weight  in  pounds  required  to  break  a  unit  of  the  material 
named  when  suspended  from  the  middle,  the  piece  being 
supported  at  each  end.  The  unit  of  material  is  a  bar  one  inch 
square  and  one  foot  long  between  the  bearings.  The  third 
column,  headed  F,  contains  the  values  of  the  several  mate- 
rials named  as  to  their  resistance  to  flexure,  as  explained  in 
Arts.  302-305,  Transverse  Strains.  These  values  of  F,  as 
constants,  are  used  in  the  rules.  The  fourth  column,  headed 
^,  contains  the  values  of  the  several  materials  named,  denot- 
ing the  elasticity  of  the  fibres,  as  explained  in  Art.  312, 
Transverse  Strains.  These  values  of  e,  as  constants,  are 
used  in  the  rules. 

The  fifth  column,  headed  ay  contains  for  the  several  ma- 
terials named  the  ratio  of  the  resistance  to  flexure  as  com- 
pared with  that  to  rupture,  and  which,  as  constants  used 
in  the  rules,  indicate  the  margin  of  safety  to  be  given  for 
each  kind  of  material.  The  figures  given  in  each  case  show 
the  smallest  possible  value  that  may  be  safely  given  to  a,  the 
factor  of  safety,  x  In  practice  it  is  generally  taken  higher 
than  the  amount  given  in  the  table.  For  example,  in  the 
table  the  value  of  B,  the  constant  for  rupture  by  transverse 
strain  for  spruce,  is  550. 

Now,  if  the  dimensions  of  a  spruce  beam  to  carry  a  given 
weight  be  computed  by  the  rules,  using  the  constant  B,  at 
550,  the  beam  will  be  of  such  a  size  that  the  given  weight 
will  just  break  it. 

But  if,  in  the  computation,  instead  of  taking  the  full 
value  of  B,  only  a  part  of  it  be  taken,  then  the  beam  will  not 
break  immediately;  and  if  the  part  taken  be  so  small  that 
its  effect  upon  the  fibres  shall  not  be  sufficient  to  strain  them 
beyond  their  limit  of  elasticity,  the  beam  will  be  capable  of 
sustaining  the  weight  for  an  indefinite  period  ;  in  this  case 
the  beam  will  be  loaded  by  what  is  termed  the  safe  weight. 
Or,  since  the  value  of  a  for  spruce  is  2-23  in  the  table,  if,  in- 
stead of  taking  B  at  550,  its  full  value,  only  the  quotient 
arising  from  a  division  of  B  by  a  be  taken — or  550  divided 
by  2-23,  which  equals  246-6 — then  the  beam  will  be  of  suffi- 
cient size  to  carry  the  load  safely.  Therefore,  while  the  con- 


THE   VARIOUS   CLASSES   OF   PRESSURES.  85 

slant  B  is  to  be  used  for  a  breaking  weight,  for  a  safe  load 

n 

the  quotient  of  -   is  to  be  used.    But,  again,  if  a  be  taken  at 

its  value  as  given  in  the  table,  the  computed  beam  will  be 
loaded  up  to  its  limit  of  safety.  So  loaded  that,  if  the  load 
be  increased  only  in  a  small  degree,  the  limit  of  safety  will 
be  passed,  and  the  beam  liable,  in  time,  to  fail  by  rupture. 

Therefore,  as  the  values  of  a,  in  the  table,  are  the  smallest 
possible,  it  is  prudent  in  practice  always  to  take  a  larger 
than  the  table  value.  For  example,  the  table  value  of 
a  for  spruce  is  2-23,  but  in  practice  let  it  be  taken  at  3 
or  4. 


97. — Resistance  to  Compression. — The  resistance  of  ma- 
terials to  the  force  of  compression  may  be  considered  in 
several  ways.  Posts  having  their  heights  less  than  ten 
times  their  least  sides  will  crush  before  bending ;  these 
belong  to  one  class :  another  class  is  that  which  com- 
prises all  posts  the  height  of  which  is  equal  to  ten  times 
their  least  sides,  or  more  than  ten  times  ;  these  will  bend 
before  crushing.  Then  there  remains  to  be  considered 
the  manner  in  which  the  pressure  is  applied  :  whether  in 
line  with  the  fibres,  or  transversely  to  them ;  and,  again, 
whether  the  pressure  tends  to  crush  the  fibres,  or  simply 
to  force  off  a  part  of  the  piece  by  splitting.  The  various 
pressures  may  be  comprised  in  the  four  classes  following, 
namely : 

ist. — When  the  pressure  is  applied  to  the  fibres  trans- 
versely. 

2d.— When  the  pressure  is  applied  to  the  fibres  longi- 
tudinally, and  so  as  to  split  off  the  part  pressed  against, 
causing  the  fibres  to  separate  by  sliding. 

3d. — When  the  pressure  is  applied  to  the  fibres  longi- 
tudinally, and  on  short  pieces. 

4th.—  When  the  pressure  is  applied  to  the  fibres  longi- 
tudinally, and  on  long  pieces. 

These  four  classes  will  now  be  considered  in  their  reg- 
ular order. 


86  CONSTRUCTION. 

98. — Compression  Transversely  to  the  Fibres. — In  this 
first  class  of  compression,  experiment  has  shown  that  the 
resistance  is  in  proportion  to  the  number  of  fibres  pressed, 
that  is,  in  proportion  to  the  area.  For  example,  if  5000 
pounds  is  required  to  crush  a  prism  with  a  base  i  inch 
square,  it  will  require  20,000  pounds  to  crush  a  prism  having 
a  base  of  2  by  2  inches,  equal  to  4  inches  area ;  because  4 
times  5000  equals  20000. 

Therefore,  if  any  given  surface  pressed  be  multiplied  by 
the  pressure  per  inch  which  the  kind  of  material  pressed 
may  be  safely  trusted  with,  the  product  will  be  the  total 
pressure  which  may  with  safety  be  put  upon  the  given  sur- 
face. Now,  the  capability  for  this  kind  of  resistance  is  given 
in  column  P,  in  Table  I.,  for  each  kind  of  material  named  in 
the  table.  Therefore,  to  find  the  limit  of  weight,  proceed 
as  follows: 

99. — The  Limit  of  Weight. — To  ascertain  what  weight 
a  post  may  be  loaded  with,  so  as  not  to  crush  the  surface 
against  which  it  presses,  we  have — 

Rule  1.— Multiply  the  area  of  the  post  in  inches  by  the 
value  of  P,  Table  I.,  and  the  product  will  be  the  weight  re- 
quired in  pounds  ;  or — 

W=AP.  (i.) 

Example. — A  post,  8  by  10  inches,  stands  upon  a  white- 
pine  girder;  the  area  equals  8  x  10  =  80  inches.  This  being 
multiplied  by  320,  the  value  of  P,  Table  I.,  set  opposite  white 
pine,  the  product,  25600,  is  the  required  weight  in  pounds. 

100. — Area  of  Post.— To  ascertain  what  area  a  post  must 
have  in  order  to  prevent  the  post,  loaded  with  a  given 
weight,  from  crushing  the  surface  against  which  it  presses, 
we  have — 

Rule  II.  — Divide  the  given  weight  in  pounds  by  the  value 
of  P,  Table  I.,  and  the  quotient  will  be  the  area  required  in 
inches  ;  or-- 


RESISTANCE   TO    RUPTURE   BY   SLIDING.  87 

Example.  —  A  post  standing  on  a  Georgia-pine  girder  is 
loaded  with  100,000  pounds:  what  must  be  its  area?  The 
weight,  100000,  divided  by  900,  the  value  of  />  Table  I.,  set 
opposite  Georgia  pine,  the  quotient,  in-  11,  is  the  required 
area  in  inches.  The  post  may  be  10  by  ii|,  or  10  by  11 
inches;  or  if  square  each  side  will  be  10-54  inches,  or  IIT'¥ 
inches  diameter  if  round. 

101.—  Rupture  by  Sliding.  —  In  this  the  second  class  of 
rupture  by  compression,  it  has  been  ascertained  by  ex- 
periment that  the  resistance  is  in  proportion  to  the  area  of 
the  surface  separated  without  regard  to  the  form  of  the  sur- 
face. Now,  in  Table  I.,  column  //,  we  have  the  ultimate 
resistance  to  this  strain  of  the  several  materials  named. 
But  to  obtain  the  safe  load  per  inch,  the  ultimate  resist- 
ance of  the  table  is  to  be  divided  by  a  factor  of  safety,  of 
such  value  as  circumstances  may  seem  to  require.  Gener- 
ally this  factor  may  be  taken  at  3.  Then  to  obtain  the  safe 
load  for  any  given  case,  we  have  but  to  multiply  the  given 
surface  by  the  ultimate  resistance,  and  divide  by  the  factor 
of  safety  ;  therefore,  proceed  as  follows  : 

102.—  Tl»e  Limit  of  Weight.—  To  ascertain  what  weight 
may  be  sustained  safely  by  the  resistance  of  a  given  area  of 
surface,  when  the  weight  tends  to  split  off  the  part  pressed 
against  by  causing,  in  case  of  fracture,  one  surface  to  slide 
on  the  other,  we  have  — 

Rule  III.—  Multiply  the  area  of  the  surface  by  the  value 
of  Hi  in  Table  I.  divide  by  the  factor  of  safety,  and  the 
quotient  will  be  the  weight  required  in  pounds  ;  or— 

W  =  -*-Z  (3-) 


Example.  —  The  foot  of  a  rafter  is  framed  into  the  end  of 
its  tie-beam,  so  that  the  uncut  substance  of  the  tie-beam  is 
1  5  inches  long  from  the  end  of  the  tie-beam  to  the  joint  of 
the  rafter;  the  tie-beam  is  of  white  pine,  and  is  6  inches 
thick:  what  amount  of  horizontal  thrust  will  this  end  of 
the  tie-beam  sustain,  without  danger  of  having  the  end  oi 


88  CONSTRUCTION. 

the  tie-beam  split  off?  Here  the  area  of  surface  that  sus- 
tains the  pressure  is  6  by  15  inches,  equal  to  90  inches. 
This  multiplied  by  480,  the  value  of  H,  set  opposite  to 
white  pine,  Table  I.,  and  divided  by  3,  as  a  factor  of  safety, 
gives  a  quotient  of  14400,  and  this  is  the  required  weight  in 
pounds. 

103.  —  Area  of  Surface.  —  To  ascertain  the  area  of  surface 
that  is  required  to  sustain  a  given  weight  safely,  when  the 
weight  tends  to  split  off  the  part  pressed  against,  by  causing, 
in  case  of  fracture,  one  'surface  to  slide  on  the  other,  we 
have— 

Rule  IV.  —  Divide  the  given  weight  in  pounds  by  the 
value  of  H,  Table  I.  ;  multiply  the  quotient  by  the  factor  of 
safety,  and  the  product  will  be  the  required  area  in  inches  ; 
or  — 


(40 


Example.  —  The  load  on  a  rafter  causes  a  horizontal  thrust 
at  its  foot  of  40,000  pounds,  tending  to  split  off  the  end  of 
the  tie-beam  :  what  must  be  the  length  of  the  tie-beam  be- 
yond the  line  where  the  foot  of  the  rafter  is  framed  into  it, 
the  tie-beam  being  of  Georgia  pine,  and  9  inches  thick  ? 
The  weight,  or  horizontal  thrust,  40000,  divided  by  840,  the 
value  of  //,  Table  I.,  set  opposite  Georgia  pine,  gives  a  quo- 
tient of  47-619,  and  this  multiplied  by  3,  as  a  factor  of  safety, 
gives  a  product  of  142-857.  This,  the  area  of  surface  in 
inches,  divided  by  9,  the  breadth  of  the  surface  strained 
(equal  to  the  thickness  of  the  tie-beam),  the  quotient,  15.87, 
is  the  length  in  inches  from  the  end  of  the  tie-beam  to  the 
rafter  joint,  say  16  inches. 

I04.-Tcnon§  and  Splices.-  A  knowledge  of  this  kind  of 
resistance  of  materials  is  useful,  also,  in  ascertaining  the 
length  of  framed  tenons,  so  as  to  prevent  the  pin,  or  key, 
with  which  they  are  fastened  from  tearing  out  ;  and,  also,  in 
cases  where  tie-beams,  or  other  timber  under  a  tensile  strain, 


CRUSHING   STRENGTH   OF   STOUT   POSTS.  89 

are  spliced,  this  rule  gives  the  length  of  the  joggle  at  each 
end  of  the  splice. 

105. — Stout  Post§.— These  comprise  the  third  class  of  ob- 
jects subject  to  compression  (Art.  97),  and  include  all  posts 
which  are  less  than  ten  diameters  high.  The  resistance  to 
compression,  in  this  class,  is  ascertained  to  be  directly  in  pro- 
portion to  the  area  of  cross-section  of  the  post. 

Now  in  Table  I.,  column  Cy  the  ultimate  resistance  to 
crushing  is  given  for  the  several  kinds  of  materials  named  ; 
from  which  the  safe  resistance  per  inch  may  be  obtained  by 
dividing  it  by  a  proper  factor  of  safety.  Having  the  safe 
resistance  per  inch,  the  resistance  of  any  given  post  may  be 
determined  by  multiplying  it  by  the  area  of  the  cross-section 
of  the  post.  Therefore,  proceed  as  follows  : 

106.— TBie  Limit  of  Weight.— To  find  the  weight  that  can 
be  safely  sustained  by  a  post,  when  the  height  of  the  post  is 
less  than  ten  times  the  diameter  if  round,  or  ten  times  the 
thickness  if  rectangular,  and  the  direction  of  the  pressure 
coinciding  with  the  axis,  we  have— 

RHle  V. — Multiply  the  area  of  the  cross-section  of  the 
post  in  inches  by  the  value  of  C,  in  Table  I. ;  divide  the  pro- 
duct by  the  factor  of  safety,  and  the  quotient  will  be  the  re- 
quired weight  in  pounds  ;  or — 

W=  ^-£.  (5.) 

Example.— A  Georgia-pine  post  is  6  feet  high,  and  in 
cross-section  8  x  12  inches:  what  weight  will  it  safely  sus- 
tain? The  height  of  this  post,  12  x  6  =  72  inches,  which  is 
less  than  lox  8  (the  size  of  the  narrowest  side)  —  80  inches  ; 
it  therefore  belongs  to  the  class  coming  under  this  rule.  The 
area  =  8  x  12  =  96  inches  ;  this  multiplied  by  9500,  the  value 
of  C,  in  the  table,  set  opposite  Georgia  pine,  and  divided  by 
6,  as  a  factor  of  safety,  the  quotient,  152000,  is  the  weight  in 
pounds  required.  It  will  be  observed  that  the  weight  would 
be  the  same  for  a  Georgia-pine  post  of  any  height  less  than 


90  CONSTRUCTION. 

10  times  8  inches  =  So  inches  =  6  feet  8  inches,  provided  its 
breadth  and  thickness  remain  the  same,  12  and  8  inches. 


107. — Area  of  Post. — To  find  the  area  of  the  cross-sec- 
tion of  a  post  to  sustain  a  given  weight  safely,  the  height  of 
the  post  being  less  than  ten  times  the  diameter  if  round,  or 
ten  times  the  least  side  if  rectangular,  the  pressure  coinciding 
with  the  axis,  we  have— 

Rule  VI. — Divide  the  given  weight  in  pounds  by  the 
value  of  C,  in  Table  I.  -,  multiply  the  quotient  by  the  factor 
of  safety,  and  the  product  will  be  the  required  area  in 
inches ;  or— 

(6.) 


Example. — A  weight  of  40,000  pounds  is  to  be  sustained 
by  a  white-pine  post  4  feet  high :  what  must  be  its  area  of 
section  to  sustain  the  weight  safely  ?  Here,  40000  divided 
by  6650,  the  value  of  C,  in  Table  I.,  set  opposite  white  pine, 
and  the  quotient  multiplied  by  6,  as  a  factor  of  safety,  the  pro- 
duct is  36 ;  this,  therefore,  is  the  required  area,  and  such  a 
post  may  be  6  x  6  inches.  To  find  the  least  side,  so  that  it 
shall  not  be  less  than  one  tenth  of  the  height,  divide  the 
height,  reduced  to  inches,  by  10,  and  make  the  least  side  to 
exceed  this  quotient.  The  area  divided  by  the  least  side  so 
determined  will  give  the  wide  side.  If,  however,  by  this 
process,  the  first  side  found  should  prove  to  be  the  greatest, 
then  the  size  of  the  post  is  to  be  found  by  Rule  IX.,  X.,  or 
XI. 

108. — Area  of  Round  Po§t§.— In  case  the  post  is  to  be 
round,  its  diameter  may  be  found  by  reference  to  the  Table 
of  Circles  in  the  Appendix,  in  the  column  of  diameters,  op- 
posite to  the  area  of  the  post  to  be  found  in  the  column  of 
areas,  or  opposite  to  the  next  nearest  area.  For  example, 
suppose  the  required  area,  as  just  found  by  the  example 
under  Rule  VI.,  is  36 :  by  reference  to  the  column  of  areas, 
35.78  is  the  nearest  to  36,  and  the  diameter  set  opposite  is 


CRUSHING  STRENGTH  OF  SLENDER  POSTS.       9! 

6.75,  which  is  a  trifle  too  small.     The  post  may  therefore  be, 
say,  6|  inches  diameter. 

109.  —  Slender  Posts.  —  When  the  height  of  a  post  is  less 
than  ten  times  its  diameter,  the  resistance  of  the  post  to 
crushing  is  approximately  in  proportion  to  its  area  of  cross- 
section.     But  when  the  height  is  equal  to  or  more  than  ten 
diameters,  the  resistance  per  square  inch  is  diminished.    The 
resistance  diminishes  as  the  height  is  increased,  the  diameter 
remaining    the    same  (Transverse  Strains,  Art.  643).      The 
strength  of  a  slender  post  consists  in  a  combination  of  the 
resistances  of  the  material  to  bending  and  to  crushing,  and 
is  represented  in  the  following  rule  : 

110.  —  The   Limit  of  Weight.  —  To   ascertain   the  weight 
that  can  be  sustained  safely  by  a  post  the  height  of  which 
is  at  least  ten  times  its  least  side  if  rectangular,  or  ten  times 
its  diameter  if  round,  the  direction  of  the  pressure  coincid- 
ing with  the  axis,  we  have  — 

Rule  VII.  —  Divide  the  height  of  the  post  in  inches  by  the 
diameter,  or  least  side,  in  inches  ;  multiply  the  quotient  by 
itself,  or  take  its  square  ;  multiply  the  square  by  the  value 
of  e,  in  Table  III.,  set  opposite  the  kind  of  material  of  which 
the  post  is  made  ;  to  the  product  add  the  half  of  itself  ;  to 
the  sum  add  unity  (or  one)  ;  multiply  this  sum  by  the  factor 
of  safety,  and  reserve  the  product  for  use,  as  below.  Now 
multiply  the  area  of  cross-section  of  the  post  in  inches  by 
the  value  of  C,  in  Table  I.,  set  opposite  the  material  of  the 
post,  and  divide  the  product  by  the  above  reserved  product; 
the  quotient  will  be  the  required  weight  in  pounds  ;  or— 


(7.) 


Example  :  A  Round  Post.—  What  weight  may  be  safely 
placed  upon  a  post  of  Georgia  pine  10  inches  diameter  and 
10  feet  high,  the  pressure  coinciding  with  the  axis  of  the 
post?  The  height  of  the  post,  dox  12  =)  120  inches,  divided 
by  10,  its  diameter,  gives  a  quotient  of  12;  this  multiplied 


92  CONSTRUCTION. 

by  itself  gives  144,  its  square;  and  this  by  -00109,  the  value 
of  e  for  Georgia  pine,  in  Table  III.,  gives  •  15696  ;  to  which 
adding  its  half,  the  sum  is  0-23544;  to  which  adding  unity, 
the  sum  is  1-23544 ;  and  this  multiplied  by  7,  as  a  factor  of 
safety,  the  product  is  8 -648,  the  reserved  divisor.  Now  the 
area  of  the  post  is  (see  Table  of  Areas  of  Circles,  in  the  Ap- 
pendix, opposite  its  diameter,  10)  78-54;  this  multiplied  by 
9500,  the  value  of  C  for  Georgia  pine,  in  Table  I.,  gives  a 
product  of  746130;  which  divided  by  8-648,  the  above  re- 
served divisor,  gives  a  quotient  of  86278,  the  required  weight 
in  pounds. 

Anotlier  Example  :  A  Rectangular  Post. — What  weight  may 
be  safely  placed  upon  a  white-pine  post  lox  12  inches,  and 
15  feet  high,  the  pressure  coinciding  with  the  axis  of  the 
post?  Proceeding  according  to  the  rule,  we  find  the  height 
of  the  post  to  be  180  inches,  which  divided  by  10,  the  least 
side  of  the  post,  gives  18  ;  this  multiplied  by  itself  gives  324* 
its  square  ;  which  multiplied  by  -0014,  the  value  of  e  for 
white  pine,  in  Table  III.,  gives  -4536;  to  which  adding  its 
half,  the  sum  is  -6804;  to  which  adding  unity,  the  sum  is 
i  -6804  ;  and  this  multiplied  by  8,  as  a  factor  of  safety,  the  pro- 
duct is  13-4432,  the  reserved  divisor.  Now  the  area  of  the 
post,  ( 10  x  12  =)  120  inches,  multiplied  by  6650,  the  value  of 
C  for  white  pine,  in  Table  I.,  gives  a  product  of  798,000,  and 
this  divided  by  13-4432,  the  above  reserved  divisor,  the  quo- 
tient, 59360,  is  the  required  weight  in  pounds. 

III. — Diameter  of  the  Post:  when  Round.— To  ascertain 
the  size  of  a  round  post  to  sustain  safely  a  given  weight, 
when  the  height  of  the  post  is  at  least  ten  times  the  diameter ; 
the  direction  of  the  pressure  coinciding  with  the  axis  of  the 
post;  we  have — 

Rule  VIII. —Multiply  the  given  weight  by  the  factor  of 
safety,  and  divide  the  product  by  1-5708  times  the  value  of  C 
for  the  material  of  the  post,  found  in  Table  I.  ;  reserve  the 
quotient,  calling  its  value  G.  Now  multiply  432  times  the 
value  of  c  for  the  material  of  the  post,  found  in  Table  III., 
by  the  square  of  the  height  in  feet,  and  by  the  above  quo- 
tient G ;  to  the  product  add  the  square  of  G :  extract  the 


SIZE   OF   POST   FOR   GIVEN   WEIGHT.  93 

square  root  of  the  sum,  and  to  it  add  the  value  of  G ;  then 
the  square  root  of  this  sum  will  be  the  required  diameter; 

or— 

r  Wa 

Cr    nzr 

,5708  L 

d    =4  /      .    /  At?    (T~f    /8   O.    f^    4-   /C   *  (90 


Example.  —  What  should  be  the  diameter  of  a  locust  post 
10  feet  high  to  sustain  safely  40,000  pounds,  the  pressure 
coinciding  with  the  axis  ?  Proceeding  by  the  rule,  the  given 
weight  multiplied  by  6,  taken  as  a  factor  of  safety,  equals 
240000.  Dividing  this  by  1-5708  times  11700,  the  value  of 
C  for  locust,  in  Table  I.,  the  quotient,  13-06,  is  the  value  of 
G,  the  square  of  which  is  170-53.  Now,  the  value  of  e  for 
locust,  in  Table  III.,  is  -0015.  This  multiplied  by  432,  by 
100,  the  square  of  the  height,  and  by  the  above  value  of  G, 
gives  a  product  of  846-2  ;  which  added  to  170-53,  the  above 
square  of  G,  gives  the  sum  of  1016-73.  To  31-89,  the  square 
root  of  this,  add  the  above  value  of  G  ;  then  6-7,  the  square 
root  of  this  sum,  is  the  required  diameter  of  the  post.  The 
post  therefore  requires  to  be  6-7,  say  6£-  inches  diameter. 

112.—  Side  of  tlic  Post:  when  Square.—  To  ascertain  the 
side  of  a  square  post  to  sustain  safely  a  given  weight,  when 
the  height  of  the  post  is  at  least  ten  times  the  side  ;  the  pres- 
sure coinciding  with  the  axis  ;  we  have  — 

Rule  IX.—  Multiply  the  given  weight  by  the  factor  of 
safety,  and  divide  the  product  by  twice  the  value  of  C  for 
the  material  of  the  post,  found  in  Table  I.  ;  reserve  the  quo- 
tient, calling  its  value  G..  Now  multiply  432  times  the  value 
of  e  for  the  material  of  the  post,  found  in  Table  III.,  by  the 
square  of  the  height  in  feet,  and  by  the  above  quotient  G\ 
to  the  product  add  the  square  of  G  ;  extract  the  square  root 
of  the  sum,  and  to  it  add  the  value  of  G  ;  then  the  square 
root  of  this  sum  will  be  the  required  side  of  the  post  ;  or— 


.  Co-) 

2  C 


94  CONSTRUCTION. 


•S  =4/  4/432  Ge  I  *  +  G'2  +  G. 


Example. — What  should  be  the  side  of  a  Georgia-pine 
square  post  1 5  feet  high  to  sustain  safely  50,000  pounds,  the 
pressure  coinciding  with  the  axis  of  the  post?  Proceeding 
by  the  rule,  50,000  pounds  multiplied  by  6,  as  a  factor  of 
safety,  gives  300000 ;  this  divided  by  2  x  9500  (the  value  of 
€)•=•  19000,  the  quotient,  15-789,  is  the  value  of  G.  The 
value  of  e  for  Georgia  pine  is  -00109;  tne  square  of  the 
height  is  225  ;  then,  432  times  -00109  by  225  and  by  15-789 
(the  above  value  of  G)  gives  a  product  of  1672  -  86  ;  the  square 
of  £  equals  249-31  ;  this  added  to  1672-86  gives  a  sum  of 
1922- 17,  the  square  root  of  which  is  43-843  ;  which  added* to 
15-789,  the  value  of  G,  gives  59-632,  the  square  root  of  which 
is  7-722,  the  required  side  of  the  post.  The  post,  therefore, 
requires  to  be,  say,  7f  inches  square. 

113. — Thickness  of  a  Rectangular  Post.— This  may  be 
definitely  ascertained  when  the  proportion  which  the  thick- 
ness shall  bear  to  the  breadth  shall  have  been  previously 
determined.  For  example,  when  the  proportion  is  as  6  to  8, 
then  i  J  times  6  equals  8,  and  the  proportion  is  as  1  to  i£; 
again,  when  the  proportion  is  as  8  to  10,  then  ij  times  8 
equals  10,  and  in  this  case  the  proportion  is  as  i  to  ij.  Let 
the  latter  figure  of  the  ratio  i  to  ij,  i  to  ij,  etc.,  be  called 
n,  or  so  that  the  proportion  shall  be  as  i  to  ny  then — 

To  ascertain  the  thickness  of  a  post  to  sustain  safely  a 
given  weight,  when  the  height  is  at  least  ten  times  the  thick- 
ness ;  the  action  of  the  weight  coinciding  with  the  axis ;  we 
have — 

Rule  X. — Multiply  the  given  weight  by  the  factor  of 
safety,  and  divide  the  product  by  twice  the  value  of  C  for 
the  material  of  the  post,  found  in  Table  I.,  multiplied  by  ;/, 
as  above  explained  ;  reserve  the  quotient,  calling  it  G.  Now 
multiply  432  times  the  value  of  e  for  the  material  of  the  post, . 
found  in  Table  III.,  by  the  square  of  the  height  in  feet,  and 
by  the  above  quotient  G ;  to  the  product  add  the  square  of 
G ;  extract  the  square  root  of  the  sum,  and  to  it  add  th£  value 


BREADTH    OF   POST   FOR   GIVEN   THICKNESS. 


95 


of  G ;  then  the  square  root  of  this  sum  will  be  the  required 
thickness  of  the  post ;  or — 

Wa 

G—-~ TT- -.  (12.) 

2  C  n  v     ' 


t  =  V  1/432  G 


(130 


Example.— What  should  be  the  thickness  of  a  white-pine 
rectangular  post  20  feet  high  to  sustain  safely  30,000  pounds, 
the  pressure  coinciding  with  the  axis,  and  the  proportion 
between  the  thickness  and  breadth  to  be  as  10  to  12,  or  as  I 
to  i  -2  ?  Proceeding  according  to  the  rule,  we  have  the  pro- 
duct of  30000,  the  given  weight,  by  6,  as  a  factor  of  safety, 
equals  180000  ;  this  divided  by  twice  Cx  n,  or  2  x  6650  x  i  -2, 
(=15960)  gives  a  quotient  of  11-278,  the  value  of  G.  Then, 
we  have  c=  -0014,  the  square  of  the  height  equals  400; 
therefore, 432  x  -0014x400x11.278  =  2728-43.  Tothisadd- 
ing  127-2,  the  square  of  £,  we  have  2855  63,  the  square 
root  of  which  is  53-438;  and  this  added  to  G  gives  64-716, 
the  square  root  of  which  is  8-045,  tne  required  thickness  of 
the  post.  Now,  since  the  thickness  is  in  proportion  to  the 
breadth  as  i  to  1-2,  therefore  8-045  x  1-2  =  9-654,  the  re- 
quired width.  The  post,  therefore,  may  be  made  8x9! 
inches. 

114-.— Breadth  of  a  Rectangular  Post.— When  the  thick- 
ness of  a  post  is  fixed,  and  the  breadth  required ;  then,  to 
ascertain  the  breadth  of  a  rectangular  post  to  sustain  safely 
a  given  weight,  the  direction  of  the  pressure  of  which  coin- 
cides with  the  axis  of  the  post,  we  have — 

Rule  XI. —  Divide  the  height  in  inches  by  the  given  thick- 
ness, and  multiply  the  quotient  by  itself,  or  take  its  square ; 
multiply  this  square  by  the  value  of  e  for  the  material  of  the 
post,  found  in  Table  111. ;  to  the  product  add  its  half,  and  to 
the  sum  add  unity  ;  multiply  this  sum  by  the  given  weight, 
and  by  the  factor  of  safety ;  divide  the  product  by  the  pro- 
duct of  the  given  thickness  multiplied  by  the  value  of  C  for 


96  CONSTRUCTION. 

the  material  of  the  post,  found  in  Table  I.,  and  the  quotient 
will  be  the  required  breadth  ;  or— 


Example.  —  What  should  be  the  breadth  of  a  spruce  post 
1  8  feet  high  and  6  inches  thick  to  sustain  safely  25,000 
pounds,  the  pressure  coinciding  with  the  axis  of  the  post  ? 
According  to  the  rule,  216  (=  12  x  18),  the  height  in  inches, 
divided  by  6,  the  given  thickness,  gives  a  quotient  of  36,  the 
square  of  which  is  1296;  the  value  of  e  for  spruce  is  -00098  ; 
this  multiplied  by  1296,  the  above  square,  equals  i  -27  ;  which 
increased  by  -635,  its  half,  amounts  to  1-905  ;  this  increased 
by  unity,  the  sum  is  2-905  ;  which  multiplied  by  the  given 
weight,  and  by  the  factor  of  safety,  gives  a  product  of 
435749;  and  this  divided  by  6  (the  given  thickness)  times  7850 
(the  value  of  C  for  spruce)  =  47  1  oo,  gives  a  quotient  of  9  •  2  5  1  6, 
the  required  breadth  of  the  post.  The  post,  therefore,  re- 
quires to  be  6  x  9^  inches. 

Observe  that  when  the  breadth  obtained  by  the  rule  is 
less  than  the  given  thickness,  the  result  shows  that  the  con- 
ditions of  the  case  are  incompatible  with  the  rule,  and  that 
a  new  computation  must  be  made  ;  taking  now  for  the 
breadth  what  was  before  understood  to  be  the  thickness, 
and  proceeding  in  this  case,  by  Rule  X.,  to  find  the  thickness. 

115.  —  Resistance  to  Ten§ion.—  In  Art.  95  are  recorded  the 
results  of  experiments  made  to  test  the  resistance  of  vari- 
ous materials  to  tensile  strain,  showing  in  each  case  the  ca- 
pability to  such  resistance  per  square  inch  of  sectional  area. 
The  action  of  materials  in  resisting  a  tensile  strain  is  quite 
simple  ;  their  resistance  is  found  to  be  directly  as  their  sec- 
tional area.     Hence  — 

116.  —  The  rim  it  of  Weight.—  To  ascertain  the  weight  or 
pressure  that  may  be  safely  applied  to  a  beam  or  rod  as  a 
tensile  strain,  we  have  — 

Rule  XII.  —  Multiply  the  area  of  the  cross-section  of  the 
beam  or  rod  in  inches  by  the  value  of  'T,  Table  II.  ;  divide 


AREA   OF   BEAM    FOR  TENSILE   STRAIN.  97 

the  product  by  the  factor  of  safety,  and  the  quotient  will  be 
the  required  weight  in  pounds ;  or — 


(150 

The  cross-section  here  intended  is  that  taken  at  the  small- 
est part  of  the  beam  or  rod.  A  beam,  in  framing,  is  usually 
cut  with  mortices ;  the  area  will  probably  be  smallest  at  the 
severest  cutting  ;  the  area  used  in  the  rule  must  be  that  of  the 
uncut  fibres  only. 

Example. — The  tie-beam  of  a  roof-truss  is  of  white  pine, 
6  x  10  inches ;  the  cutting  for  the  foot  of  the  rafter  reduces 
the  uncut  area  to  40  inches  :  what  amount  of  horizontal  thrust 
from  the  foot  of  the  rafter  will  this  tie-beam  safely  sustain  ? 
Here  40  times  12000,  the  value  of  T,  equals  480000;  this 
divided  by  6,  as  a  factor  of  safety,  gives  80000,  the  required 
weight  in  pounds. 

(17. — Sectional  Area. — To  ascertain  the  sectional  area  of 
a  beam  or  rod  that  will  sustain  a  given  weight  safely,  when 
applied  as  a  tensile  strain,  we  have — 

Rule  XIII. — Multiply  the  given  weight  in  pounds  by  the 
factor  of  safety  ;  divide  the  product  by  the  value  of  T,  Table 
II.,  and  the  quotient  will  be  the  area  required  in  inches; 
or — 

A=^.  (16.) 

This  is  the  area  of  uncut  fibres.  If  the  piece  is  to  be  cut 
for  mortices,  or  for  any  other  purpose,  then  for  this  an 
adequate  addition  is  to  be  made  to  the  result  found  by  the 
rule. 

Example. — A  rafter  produces  a  thrust  horizontally  of  80,000 
pounds ;  the  tie-beam  is  to  be  of  oak  :  what  must  be  the 
area  of  the  cross-section  of  the  tie-beam  in  order  to  sustain 
the  rafter  safely  ?  The  given  weight,  80000,  multiplied  by 
10,  as  a  factor  of  safety,  gives  800000;  this  divided  by  19500, 
the  value  of  7",  Table  II.,  the  quotient,  41,  is  the  area  of  uncut 
fibres.  This  should  have  usually  one  half  of  its  amount 


98  CONSTRUCTION. 

added  to  it  as  an  allowance  for  cutting;  therefore,  41+21 
=  62.  The  tie-beam  may  be  6  x  loj  inches. 

Another  Example.  —  A  tie-rod  of  American  refined 
wrought  iron  is  required  to  sustain  safely  36,000  pounds  : 
what  should  be  its  area  of  cross-section  ?  Taking  7  as  the 
factor  of  safety,  7x36000=  252000;  and  this  divided  by 
60000,  the  value  of  7",  Table  II.,  gives  a  quotient  of  4-  2  inches, 
the  required  area  of  the  rod. 

118.  —  Weight  of  the  Suspending  Piece  Included.—  Pieces 

subjected  to  a  tensile  strain  are  frequently  suspended  verti- 
cally. In  this  case,  at  the  upper  end,  ,the  strain  is  due  not 
only  to  the  weight  attached  at  the  lower  end,  but  also  to  the 
weight  of  the  rod  itself.  Usually,  in  timber,  this  is  small 
in  comparison  with  the  load,  and  may  be  neglected  ; 
although  in  very  long  timbers,  and  where  accuracy  is  decid- 
edly essential,  as,  also,  when  the  rod  is  of  iron,  it  may  form 
a  part  of  the  rule.  Taking  the  effect  of  the  weight  of  the 
beam  into  account,  the  relation  existing  between  the  weights 
and  the  beam  requires  that  the  rule  for  the  weight  should 
be  as  follows  : 

Rule  XIV.  —  Divide  the  value  of  T  for  the  material  of  the 
beam  or  rod,  Table  II.,  by  the  factor  of  safety  ;  from  the 
quotient  subtract  0-434  times  the  specific  gravity  of  the  ma- 
terial in  the  beam  or  rod  multiplied  by  the  length  of  the 
beam  or  rod  in  feet  ;  multiply  the  remainder  by  the  area  of 
cross-section  in  inches,  and  the  product  will  be  the  required 
weight  in  pounds  ;  or  — 


W=A  -0-434 

N.  B.  —  This  rule  is  based  upon  the  condition  that  the  sus- 
pending piece  be  not  cut  by  mortices  or  in  any  other  way. 

Example.  —  What  weight  may  be  safely  sustained  by  a 
white-pine  rod  4x6  inches,  40  feet  long,  suspended  verti- 
cally? For  white  pine  the  value  of  T  is  12000;  this  divid- 
ed by  8,  as  a  factor  of  safety,  gives  1  500  ;  from  which  sub- 
tracting 0-434  times  0-458  (the  specific  gravity  of  white  pine, 
Table  II.)  multiplied  by  40,  the  length  in  feet,  the  remainder 


RESISTANCE   TO   TRANSVERSE    STRAINS.  99 

is  1492-049;  which  multiplied  by  24  (  =  4x6,  the  area  of 
cross-section)  equals  35,761  pounds,  the  required  weight  to  be 
carried.  The  weight  which  the  rule  would  give,  neglecting 
the  weight  of  the  rod,  would  have  been  36000;  ordinarily, 
so  slight  a  difference  would  be  quite  unimportant. 

119. — Area  of  Suspending  Piece. — To  ascertain  the  area 
of  a  suspended  rod  to  sustain  safely  a  given. weight,  when 
the^  weight  of  the  suspending  piece  is  regarded,  we  have — 

Rule  XV. — Multiply  0-434  times  the  specific  gravity  of 
the  suspending  piece  by  the  length  in  feet ;  deduct  the  pro- 
duct from  the  quotient  arising  from  a  division  of  the  value 
of  T,  Table  II.,  by  the  factor  of  safety,  and  with  the  remain- 
der divide  the  given  weight  in  pounds ;  the  quotient  will  be 
the  required  area  in  inches  ;  or — 


A  = 


T  ,  '  (18.) 

—  —  o-434/.y 

a' 


N.B. — This  rule  is  based  upon  the  condition  that  the  rod 
be  not  injured  in  anywise  by  cutting. 

Example. — What  should  be  the  area  of  a  bar  of  English 
cast  iron  20  feet  long  to  sustain  safely,  suspended  from  its 
lower  end,  a  weight  of  5000  pounds  ?  Taking  the  factor  of 
safety  at  7-0,  and  the  specific  gravity  also  at  7,  and  the 
value  of  T,  Table  II.,  at  17000,  we  have  the  product  of 
0-434  x  7-0  x  20  =  60-76;  then  17000  divided  by  7  gives 
a  quotient  of  2428-57;  from  which  deducting  the  above 
60-76,  there  remains  2367-81  ;  dividing  5000,  the  given 
weight,  by  this  remainder,  we  have  the  quotient,  2-11,  which 
is  the  required  area  in  inches. 

RESISTANCE    TO   TRANSVERSE   STRAINS. 

120. Tran§ver§e    Strains:    Rupture.— A    load    placed 

upon  a  beam,  laid  horizontally  or  inclined,  will  bend  it,  and, 
if  the  weight  be  proportionally  large,  will  break  it.  The 
power  in  the  material  that  resists  this  bending  or  breaking 
is  termed  the  resistance  to  cross-strains,  or  transverse  strains. 


100  CONSTRUCTION. 

While  in  posts  or  struts  the  material  is  compressed  or  short- 
ened, and  in  ties  and  suspending  pieces  it  is  extended  or 
lengthened,  in  beams  subjected  to  cross-strains  the  material 
is  both  compressed  and  extended.  (See  Art.  91.)  When  the 
beam  is  bent  the  fibres  on  the  concave  side  are  compressed, 
while  those  on  the  convex  side  are  extended.  The  line 
where  these  two  portions  of  the  beam  meet — that  is,  the 
portion  compressed  and  the  portion  extended  —  the  hori- 
zontal line  of  juncture,  is  termed  the  neutral  line  or  plane. 
It  is  so  called  because  at  this  line  the  fibres  are  neither  com- 
pressed nor  extended,  and  hence  are  under  no  strain  what- 
ever. The  location  of  this  line  or  plane  is  not  far  from  the 
middle  of  the  depth  of  the  beam,  when  the  strain  is  not  suf- 
ficient to  injure  the  elasticity  of  the  material ;  but  it  re- 
moves towards  the  concave  or  convex  side  of  the  beam  as 
the  strain  is  increased,  until,  at  the  period  of  rupture,  its 
distance  from  the  top  of  the  beam  is  in  proportion  to  its  dis- 
tance from  the  bottom  of  the  beam  as  the  tensile  strength  of 
the  material  is  to  its  compressive  strength. 

121. — Location  of  Mortices. — In  order  that  the  diminution 
of  the  strength  of  a  beam  by  framing  be  as  small  as  possible, 
all  mortices  should  be  located  at  or  near  the  middle  of  the 
depth.  There  is  a  prevalent  idea  with  some,  who  are  aware 
that  the  upper  fibres  of  a  beam  are  compressed  when  sub- 
ject to  cross-strains,  that  it  is  not  injurious  to  cut  these  top 
fibres,  provided  that  the  cutting  be  for  the  insertion  of  an- 
other piece  of  timber — as  in  the  case  of  gaining  the  ends  of 
beams  into  the  side  of  a  girder.  They  suppose  that  the  piece 
filled  in  will  as  effectually  resist  the  compression  as  the  part 
removed  would  have  done,  had  it  not  been  taken  out.  Now, 
besides  the  effect  of  shrinkage,  which  of  itself  is  quite  suf- 
ficient to  prevent  the  proper  resistance  to  the  strain,  there 
is  the  mechanical  difficulty  of  fitting  the  joints  perfectly 
throughout ;  and,  also,  a  great  loss  in  the  power  of  resist- 
ance, as  the  material  is  so  much  less  capable  of  resistance 
when  pressed  at  right  angles  to  the  direction  of  the  fibres 
than  when  directly  with  them,  as  the  results  of  the  experi- 
ments in  the  tables  show. 


STRENGTH   OF   BEAMS   FOR   CROSS-STRAINS.  IOI 

122. — Transverse  Strains  :  Relation  of  Weight  to  Di- 
mensions.— The  strength  of  various  materials,  in  their  re- 
sistance to  cross-strains,  is  given  in  Table  III.,  Art.  96.  The 
second  column  of  the  table  contains  the  results  of  experi- 
ments made  to  test  their  resistance  to  rupture.  In  the  case 
of  each  material,  the  figures  given  and  represented  by  B 
indicate  the  pounds  at  the  middle  required  to  break  a  unit 
of  the  material,  or  a  piece  i  inch  square  and  i  foot  long 
between  the  bearings  upon  which  the  piece  rests.  To  be 
able  to  use  these  indices  of  strength,  in  the  computation  of 
the  strength  of  large  beams,  it  is  requisite,  first,  to  establish 
the  relation  between  the  unit  of  material  and  the  larger 
beam.  Now,  it  may  be  easily  comprehended  that  the  strength 
of  beams  will  be  in  proportion  to  their  breadth ;  that  is, 
when  the  length  and  depth  remain  the  same,  the  strength 
will  be  directly  as  the  breadth  ;  lor  it  is  evident  that  a  beam 
2  inches  broad  will  bear  twice  as  much  as  one  which  is  only 
i  inch  broad,  or  that  one  which  is  6  inches  broad  will  bear 
three  times  as  much  as  one  which  is  2  inches  broad.  This 
establishes  the  relation  of  the  weight  to  the  breadth.  With 
the  depth,  however,  the  relation  is  different ;  the  strength  is 
greater  than  simply  in  proportion  to  the  depth.  If  the 
boards  cut  from  a  squared  piece  of  timber  be  piled  up  in 
the  order  in  which  they  came  from  the  timber,  and  be  loaded 
with  a  heavy  weight  at  the  middle,  the  boards  will  deflect 
or  sag  much  more  than  they  would  have  done  in  the  timber 
before  sawing.  The  greater  strength  of  the  material  when 
in  a  solid  piece  of  timber  is  due  to  the  cohesion  of  the  fibres 
at  the  line  of  separation,  by  which  the  several  boards,  before 
sawing,  are  prevented  from  sliding  upon  each  other,  and 
thus  the  resistance  to  compression  and  tension  is  made  to 
contribute  to  the  strength.  This  resistance  is  found  to  be 
in  proportion  to  the  depth.  Thus  the  strength  due  to  the 
depth  is,  first,  that  which  arises  from  the  quantity  of  the 
material  (the  greater  the  depth,  the  more  the  material), 
which  is  in  proportion  to  the  depth  ;  then,  that  which  en- 
sues from  the  cohesion  of  the  fibres  in  such  a  manner  as  to 
prevent  sliding  ;  this  is  also  as  the  depth.  Combining  the 
two,  we  have,  as  the  total  result,  the  resistance  in  proportion 


102  CONSTRUCTION. 

to  the  square  of  the  depth.  The  relation  between  the 
weight  and  the  length  is  such  that  the  longer  the  beam  is, 
the  less  it  will  resist  ;  a  beam  which  is  20  feet  long  will  sus- 
tain only  half  as  much  as  one  which  is  10  feet  long  ;  the 
breadth  and  depth  each  being  the  same  in  the  two  beams. 
From  this  it  results  that  the  resistance  is  inversely  in  pro- 
portion to  the  length.  To  obtain,  therefore,  the  relation 
between  the  strength  of  the  unit  of  material  and  that  of  a 
larger  beam,  we  have  these  facts,  namely  :  the  strength  of 
the  unit  is  the  value  of  B,  as  recorded  in  Table  III.  ;  and 
the  strength  of  the  larger  beam,  represented  .  by  W,  the 
weight  required  to  break  it,  is  the  product  of  the  breadth 
into  the  square  of  the  depth,  divided  by  the  length  ;  or, 
while  for  the  unit  we  have  the  ratio  — 

B\  i, 
we  have  for  the  larger  beam  the  ratio  — 


Therefore,  putting  these  ratios  in  an  expressed  proportion, 
we  have  — 


From  which  (the  product  of  the  means  equalling  the  pro- 
duct of  the  extremes  ;  see  Art.  373)  we  have— 


w  = 


In  which  W  represents  the  pounds  required  to  break  a 
beam,  when  acting  at  the  middle  between  the  two  supports 
upon  which  the  beam  is  laid  ;  of  which  beam  b  represents 
the  breadth  and  d  the  depth,  both  in  inches,  and  /  the  length 
in  feet  between  the  supports  ;  and  B  is  from  Table  III.,  and 
represents  the  pounds  required  to  break  a  unit  of  material 
like  that  contained  in  the  larger  beam. 


fuinvERsm 

LIMIT   OF   WEIGHT   AT   MIDDLE.  ?&&,>' 

123.— Safe  Weight:  Load  at  Middle.— The  relation 
established,  in  the  last  article,  between  the  weight  and  the 
dimensions  is  that  which  exists  at  the  moment  of  rupture. 
The  rule  (19.)  derived  therefrom  is  not,  therefore,  directly 
practicable  for  computing  the  dimensions  of  beams  for 
buildings.  From  it,  however,  one  may  readily  be  deduced 
which  shall  be  practicable.  In  the  fifth  column  of  Table  III. 
are  given  the  least  values  of  a,  the  factor  of  safety,  explained 
in  Art.  96.  Now,  if  in  place  of  By  the  symbol  for  the  break- 
ing weight,  the  quotient  of  B  divided  by  a  be  substituted, 
then  the  rule  at  once  becomes  practicable  ;  the  results  now 
being  in  consonance  with  the  requirements  for  materials 
used  in  buildings.  Thus,  with  this  modification,  we  have — 


Therefore,  to  ascertain  the  weight  which  a  beam  may  be 
safely  loaded  with  at  the  centre,  we  have— 

Rule  XVI.— Multiply  the  value  of  B,  Table  HI.,  for  the 
kind  of  material  in  the  beam  by  the  breadth  and  by  the 
square  of  the  depth  of  the  beam  in  inches ;  divide  the  pro- 
duct by  the  product  of  the  factor  of  safety  into  the  length 
of  the  beam  between  bearings  in  feet,  and  the  quotient  will 
be  the  weight  in  pounds  that  the  beam  will  safely  sustain 
at  the  middle  of  its  length. 

Example. —  What  weight  in  pounds  can  be  suspended 
safely  from  the  middle  of  a  Georgia-pine  beam  4x  10  inches, 
and  20  feet  long  between  the  bearings  ?  For  Georgia  pine  the 
value  of  B,  in  Table  III.,  is  850,  and  the  least  value  of  a  is 
1-84.  For  reasons  given  in  Art.  96,  let  a  be  taken  as  high 
as  4;  then,  in  this  case,  the  value  of  b  is  4,  and  that  of  d  is 
10,' while  that  of/  is  20.  Therefore,  proceeding  by  the  rule, 
850  x  4  x  io2  =  340000  ;  this  divided  by  4  x  20  (—  80)  gives 
a  quotient  of  4250  pounds,  the  required  weight. 

Observe  that,  had  the  value  of  a  been  taken  at  3,  instead 
of  4,  the  result  by  the  rule  would  have  been  a  load  of  5667 
pounds,  instead  of  4250,  and  the  larger  amount  would  be 
none  too  much  for  a  safe  load  upon  such  a  beam  ;  although, 


104  CONSTRUCTION. 

with  it,  the  deflection  would  be  one  third  greater  than  with 
the  lesser  load.  The  value  of  a  should  always  be  assigned 
higher  than  the  figures  of  the  table,  which  show  it  at  its 
least  value  ;  but  just  how  much  higher  must  depend  upon 
the  firmness  required  and  the  conditions  of  each  particular 
case. 

124.  —  Breadth  of  Beam  willi  Safe  Load.  —  By   a   simple 
transposition  of  the  factors  in  equation  (20.),  we  obtain  — 


a  rule  for  the  breadth  of  the  beam. 

Therefore,  to  ascertain  what  should  be  the  breadth  of  a 
beam  of  given  depth  and  length  to  safely  sustain  at  the 
middle  a  given  weight,  we  have  — 

Rule  XVII.  —  Multiply  the  given  weight  in  pounds  by 
the  factor  of  safety,  and  by  the  length  in  feet,  and  divide  the 
product  by  the  square  of  the  depth  multiplied  by  the  value 
of  B  for  the  material  in  the  beam,  in  Table  III.  ;  the  quotient 
will  be  the  required  breadth. 

Example.  —  What  should  be  the  breadth  of  a  white-pine 
beam  8  inches  deep  and  10  feet  long  between  bearings  to 
sustain  safely  2400  pounds  at  the  middle  ?  For  white  pine 
the  value  of  B,  in  Table  III.,  is  500.  Taking  the  value  of  a 
at  4,  and  proceeding  by  the  rule,  we  have  2400  x4x  10  = 
96000  ;  this  divided  by  (8a  x  500  =)  32000  gives  a  quotient 
of  3,  the  required  breadth  of  the  beam. 

125.  —  Depth  of  Beam  with  Safe  Load.  —  A  transposition 
of  the  factors  in  equation  (21.),  and  marking  it  for  extraction 
of  the  square  root,  gives  — 


a  rule  for  the  depth  of  a  beam.  Therefore,  to  ascertain  what 
should  be  the  depth  of  a  beam  of  given  breadth  and  length 
to  safely  sustain  a  given  weight  at  the  middle,  we  have — 


WEIGHT   AT   ANY   POINT. 


105 


Rule  XVIIL— Multiply  the  given  weight  by  the  factor  of 
safety,  and  by  the  length  in  feet ;  divide  the  product  by  the 
product  of  the  breadth  into  the  value  of  B  for  the  kind  of 
wood,  Table  III. ;  then  the  square  root  of  the  quotient  will 
be  the  required  depth. 

Example. — What  should  be  the  depth  of  a  spruce  beam 
5  inches  broad  and  lofeet  long  between  bearings  to  sustain 
safely,  at  middle,  4500  pounds  ?  The  value  of  B  from  the  table 
is  550;  taking  a  at  4,  and  proceeding  by  the  rule,  we  have 
4500  x  4  x  15  =  270000;  this  divided  by  (550  x  5  =)  2750 
gives  a  quotient  of  98-18,  the  square  root  of  which  is  9-909, 
the  required  depth  of  the  beam.  The  beam  should  be  5  x  10 
inches. 

126. — Safe  Load  at  any  Point. — When  the  load  is  at  the 
middle  of  a  beam  it  exerts  the  greatest  possible  strain  ;  at 
any  other  point  the  strain  would  be  less.  The  strain  de- 
creases gradually  as  it  approaches  one  of  the  bearings,  and 
when  arrived  at  the  bearing  its  effect  upon  the  beam  as  a 
cross-strain  is  zero.  The  effect  of  a  weight  upon  a  beam  is 
in  proportion  to  its  distance  from  one  of  the  bearings,  mul- 
tiplied by  the  portion  of  the  load  borne  by  that  bearing. 

The  load  upon  a  beam  is  divided  upon  the  two  bearings, 
as  shown  at  Art.  88.  The  weight  which  is  required  to  rup- 
ture a  beam  is  in  proportion  to  the  breadth  and  square  of 
the  depth,  b  d*,  as  before  shown,  and  also  in  proportion  to 
the  length  divided  by  4  times  the  rectangle  of  the  two  parts 

into  which  the  load  divides  the  length,  or (see  Fig.  35). 

4  MI  11 

This,    when   the   load    is   at   the    middle,    may    be    put    as 

=  i   a  result  coinciding  with  the  relation   before 

4x|./xi/      / 

given  in  Art.   122,  viz.  :  "The  resistance  is  inversely  in  pro- 
portion to  the  length."     The  total  resistance,  therefore,  put- 

b  d*  I 
ting  the  two  statements  together,  is  in  proportion  to  - 

A   ///  /£ 

7727 

There  are,  therefore,  these  two  ratios,  viz.,    W \  -      -  and 

4  m  i* 

B  :  I,  from  which  we  have  the  proportion— 


106  CONSTRUCTION. 


4  m  n 
from  which  we  have  — 


4  ;«  n 


(23.} 


r> 

This  is  the  relation  at  the   point  of  rupture,  and  when  —  is 

used  instead  of  B,  the    expression    gives  the    safe  weight. 
Therefore  — 

(24.) 


4  a  m  n 

is  an  expression  for  the  safe  weight.  Now,  to  ascertain  the 
weight  which  may  be  safely  borne  by  a  beam  at  any  point 
in  its  length,  we  have  — 

Rule  XIX.  —  Multiply  the  breadth  by  the  square  of  the 
depth,  by  the  length  in  feet,  and  by  the  value  of  B  for  the 
material  of  the  beam,  in  Table  III.  ;  divide  the  product  by 
the  product  of  four  times  the  factor  of  safety  into  the  rec- 
tangle of  the  two  parts  into  which  the  centre  of  gravity  of 
the  weight  divides  the  beam,  and  the  quotient  will  be  the 
required  weight  in  pounds. 

Example.  —  What  weight  may  be  safely  sustained  at  3  feet 
from  one  end  of  a  Georgia-pine  beam  which  is  4  x  10  inches, 
and  20  feet  long?  The  value  of  B  for  Georgia  pine,  in 
Table  III.,  is  850  ;  therefore,  by  the  rule,  4  x  ioa  x  20  x  850  = 
6800000.  Taking  the  factor  of  safety  at  4,  we  have 
4x4x3x17=816.  Using  this  as  a  divisor  with  which  to 
divide  the  former  product,  we  have  as  a  quotient  8333 
pounds,  the  required  weight. 

127.  —  Breadth   or   Depth:    Load   at    any    Point.  —  By    a 

proper  transposition  of  the  factors  of  (24.)  we  obtain  — 

,    ,a       4  W  a  in  11 


an  expression  showing-  the  product  of  the  breadth  into  the 
square  of  the  depth  ;  hence,  to   ascertain   the   breadth  or 


DISTRIBUTED   WEIGHT.  IO/ 

depth  of  a  beam  to  sustain  safely  a  given  weight  located  at 
any  point  on  the  beam,  we  have — 

Rule  XX. — Multiply  four  times  the  given  weight  by  the 
factor  of  safety,  and  by  the  rectangle  of  the  two  parts  into 
which  the  load  divides  the  length ;  divide  the  product  by 
the  product  of  the  length  into  the  value  of  B  for  the  mate- 
rial of  the  beam,  found  in  Table  III.,  and  the  quotient  will  be 
equal  to  the  product  of  the  breadth  into  the  square  of  the 
depth.  Now,  to  obtain  the  breadth,  divide  this  product  by 
the  square  of  the  depth,  and  the  quotient  will  be  the  required 
breadth.  But  if,  instead  of  the  breadth,  the  depth  be  de- 
sired, divide  the  said  product  by  the  breadth  ;  then  the 
square  root  of  the  quotient  will  be  the  required  depth. 

Example. — What  should  be  the  breadth  (the  depth  being 
8)  of  a  white-pine  beam  12  feet  long  to  safely  sustain  3500 
pounds  at  3  feet  from  one  end  ?  Also,  what  should  oe  its 
depth  when  the  breadth  is  3  inches?  By  the  rule,  taking 
the  factor  of  safety  at  4,  4  x  3500  x  4  x  3  x  9  =  1512000. 
The  value  of  B  for  white  pine,  in  Table  III.,  is  500 ;  there- 
fore, 500  x  12  =  6000;  with  this  as  divisor,  dividing  1512000, 
the  quotient  is  252.  Now,  to  obtain  the  breadth  when  the 
depth  is  8,  252  divided  by  (8  x  8  =)  64  gives  a  quotient  of 
3-9375,  the  required  breadth  ;  or  the  beam  may  be,  say,  4  x  8. 
Again,  when  the  breadth  is  3  inches,  we  have  for  the  quotient 
of  252  divided  by  3  =  84,  and  the  square  root  of  84  is  9- 165, 
or  9^  inches.  For  this  case,  therefore,  the  beam  should  be, 
say,  3  x  9  J  inches. 

128. — Weight  Uniformly  Distributed.— When  the  load  is 
spread  out  uniformly  over  the  length  of  a  beam,  the  beam 
will  require  just  twice  the  weight  to  break  it  that  would  be 
required  if  the  weight  were  concentrated  at  the  centre. 

Therefore,  we  have  W  —  —,  where  U  represents  the  dis- 
tributed load.  Substituting  this  value  of  W  in  equation 
(20.),  we  have — 

U__Bbd\ 

2   ~~    '  a  I   ' 

U=2-***  (26.)    - 

a  I 


IOS  CONSTRUCTION. 

Therefore,  to  ascertain  the  weight  which  may  be  safely  sus- 
tained, when  uniformly  distributed  over  the  length  of  a 
beam,  we  have  — 

Rule  XXI.  —  Multiply  twice  the  breadth  by  the  square  of 
the  depth,  and  by  the  value  of  B  for  the  material  of  the 
beam,  in  Table  III.,  and  divide  the  product  by  the  product 
of  the  length  in  feet  by  the  factor  of  safety,  and  the  quotient 
will  be  the  required  weight  in  pounds. 

Example.  —  What  weight  uniformly  distributed  may  be 
safely  sustained  upon  a  hemlock  beam  4x9  inches,  and  20 
feet  long?  The  value  of  B  for  hemlock,  in  Table  III.,  is 
450  ;  therefore,  by  the  rule,  2  x  4  x  9'  x  450  =  291600.  Tak- 
ing the  factor  of  safety  at  4,  we  have  4  x  20  —  80,  the  pro- 
duct by  which  the  former  product  is  to  be  divided.  This 
division  produces  a  quotient  of  3645,  the  required  weight. 

129.—  Breadth  or  Depth  :  Load  Uniformly  Distributed.— 

By  a  proper  transposition  of  factors  in  (26.),  we  obtain  — 


an  expression  giving  the  value  of  the  breadth  into  the  square 
of  the  depth.  From  this,  therefore,  to  ascertain  the  breadth 
or  the  depth  of  a  beam  to  sustain  safely  a  given  weight  uni- 
formly distributed  over  the  length  of  a  beam,  we  have  — 

Rule  XXII.  —  Multiply  the  given  weight  by  the  factor  of 
safety,  and  by  the  length  ;  divide  the  product  by  the  pro- 
duct of  twice  the  value  of  B  for  the  material  of  the  beam, 
in  Table  III.,  and  the  quotient  will  be  equal  to  the  breadth 
into  the  square  of  the  depth.  Now,  to  find  the  breadth, 
divide  the  said  quotient  by  the  square  of  the  depth  ;  but  if, 
instead  of  the  breadth,  the  depth  be  required,  then  divide 
said  quotient  by  the  breadth,  and  the  square  root  of  this 
quotient  will  be  the  required  depth. 

Example.  —  What  should  be  the  size  of  a  white-pine  beam  20 
feet  long  to  sustain  safely  10,000  pounds  uniformly  distributed 
over  its  length  ?  The  value  of  B  for  white  pine,  in  Table  III., 
is  500.  Let  the  factor  of  safety  be  taken  at  4.  Then,  by  the 
rule,  loooo  x  4  x  20  =  800000  ;  this  divided  by  (2  x  500  =) 


WEIGHT   PER   BEAM    IN   FLOORS.  109 

1000  gives  a  quotient  of  800.  Now,  if  the  depth  be  fixed  at 
12,  then  the  said  quotient,  800,  divided  by  (12  x  12=)  144 
gives  5-5-,  the  required  breadth  of  beam ;  and  the  beam  may 
be,  say,  5!  x  12.  Again,  if  the  breadth  is  fixed,  say,  at  6,  and 
the  depth  is  required,  then  the  said  quotient,  8co,  divided  by 
6  gives  133^,  the  square  root  of  which,  1 1  -  55,  is  the  required 
depth.  The  beam  in  this  case  should  therefore  be,  say, 
6  x  I  if  inches. 

130. — [Load  per  Foot  Superficial. — When  several^beams 
are  laid  in  a  tier,  placed  at  equal  distances  apart,  as  in  a  tier 
of  floor-beams,  it  is  desirable  to  know  what  should  be  their 
size  in  order  to  sustain  a  load  equally  distributed  over  the 
floor. 

If  the  distance  apart  at  which  they  are  placed,  measured 
from  the  centres  of  the  beams,  be  multiplied  by  the  length 
of  the  beams  between  bearings,  the  product  will  equal  the 
area  of  the  floor  sustained  by  one  beam  ;  and  if  this  area  be 
multiplied  by  the  weight  upon  a  superficial  foot  of  the  floor, 
the  product  will  equal  the  total  load  uniformly  distributed 
over  the  length  of  the  beam  ;  or,  if  c  be  put  to  represent  the 
distance  apart  between  the  centres  of  the  beams  in  feet,  and 
/  represent  the  length  in  feet  of  the  beam  between  bearings, 
and/  equal  the  pounds  per  superficial  foot  on  the  floor, 
then  the  product  of  these,  or  c  f  I,  will  represent  the  uni- 
formly distributed  load  on  a  beam ;  but  this  load  was  before 
represented  by  U  (Art.  128);  therefore,  we  have  cfl=  U, 
and  they  may  be  substituted  for  it  in  (26.)  and  (27.).  Thus 
we  have — 

b  d*  =  cflal 

or — 

(28.) 


Therefore,   to  ascertain  the  size  of  floor-beams  to  sustain 
safely  a  given  load  per  superficial  foot,  we  have— 

Rule  XXIII.— Multiply  the  given  weight  per  superficial 
foot  by  the  factor  of  safety,  by  the  distance  between  the 


HO  CONSTRUCTION. 

centres  of  the  beams  in  feet,  and  by  the  square  of  the  length 
in  feet;  divide  the  product  by  twice  the  value  of  B  for  the 
material  of  the  beams,  in  Table  III.,  and  the  quotient  will  be 
equal  to  the  breadth  into  the  square  of  the  depth.  Now,  to 
obtain  the  breadth,  divide  said  quotient  by  the  square  of  the 
depth,  and  this  quotient  will  be  the  required  breadth.  But 
if,  instead  of  the  breadth,  the  depth  be  required,  divide  the 
aforesaid  quotient  by  the  breadth  ;  then  the  square  root  of 
this  quotient  will  be  the  required  depth. 

Example. — What  should  be  the  size  of  white-pine  floor- 
beams  20  feet  long,  placed  i#  inches  from  centres,  to  sustain 
safely  90  pounds  per  superficial  foot,  including  the  weight 
of  the  materials  of  construction — the  beams,  flooring,  plas- 
tering, etc.  ?  The  value  of  B  for  white  pine  is  500 ;  the 
factor  of  safety  may  be  put  at  5.  Then,  by  the  rule,  we 
have  90  x  5  x  -ff  x  2O2  =  240000.  This  divided  by  (2  x  500 
=)  1000  gives  240.  Now,  for  the  breadth,  if  the  depth  be 
fixed  at  9  inches,  then  240  divided  by  (9*  =  )  8 1  gives  a 
quotient  of  2-963.  The  beams  therefore  should  be,  say, 
3x9.  But  if  the  breadth  be  fixed,  say,  at  2-5  inches,  then 
240  divided  by  2-5  gives  a  quotient  of  96,  the  square  root  of 
which  is  9-8  nearly.  The  beams  in  this  case  would  require 
therefore  to  be,  say,  2^  x  10  inches. 

N.  B. — It  is  well  to  observe  that  the  question  decided 
by  Rule  XXII.  is  simply  that  of  strcngtli  only.  Floor-beams 
computed  by  it  will  be  quite  safe  against  rupture,  but  they 
will  in  most  cases  deflect  much  more  than  would  be  consist- 
ent with  their  good  appearance.  Floor-beams  should  be 
computed  by  the  rules  which  include  the  effect  of  deflection. 
(See  Art.  152.) 

131. — Levers:  Load  at  One  End.— The  beams  so  far  con- 
sidered as  being  exposed  to  transverse  strains  have  been 
supposed  to  be  supported  at  each  end.  When  a  piece  is 
held  firmly  at  one  end  only,  and  loaded  at  the  other,  it  is 
termed  a  lever ;  and  the  load  which  a  piece  so  held  and 
loaded  will  sustain  is  equal  to  one  fourth  that  which  the 
same  piece  would  sustain  if  it  were  supported  at  each  end 
and  loaded  at  the  middle.  Or,  the  strain  in  a  beam  sup- 


LEVERS   TO   SUSTAIN   GIVEN   WEIGHTS.  Ill 

ported  at  each  end  caused  by  a  given  weight  located  at  the 
middle  is  equal  to  that  in  a  lever  of  the  same  breadth  and 
depth,  when  the  length  of  the  latter  is  equal  to  on6  half  that 
of  the  beam,  and  the  load  at  its  end  is  equal  to  one  half  of 
that  at  the  middle  of  the  beam.  Or,  when  P  represents  the 
load  at  the  end  of  the  lever,  and  n  its  length,  then  W—2P, 
and  l—2n.  Substituting  these  values  of  W  and  /  in  equa- 
tion (20.),  we  have  — 


4 

2an 


from  which  — 

p-Bbd* 

T^P 

Hence,  to  ascertain  the  weight  which  may  be  safely  sus- 
tained at  the  end  of  a  lever,  we  have  — 

Rule  XXIV.—  Multiply  the  breadth  of  the  lever  by  the 
square  of  its  depth,  and  by  the  value  of  B  for  the  material 
of  the  lever,  in  Table  III.  ;  divide  the  product  by  the  pro- 
duct of  four  times  the  length  in  feet  into  the  factor  of  safety, 
and  the  quotient  will  be  the  required  weight  in  pounds. 

Example.  —  What  weight  can  be  safely  sustained  at  the 
end  of  a  maple  lever  of  which  the  breadth  is  2  inches,  the 
depth  is  4  inches,  and  the  length  is  6  feet  ?  The  value  of  B 
for  maple,  in  Table  III.,  is  uoo;  therefore,  by  the  rule, 
2  x42  x  i  TOO  =  35200.  And,  taking  the  factor  of  safety  at  5, 
4x5x6=  1  20,  and  35200  divided  by  120  gives  a  quotient  of 
293  '33>  or  293^-  pounds. 

N.  B.  —  When  a  lever  is  loaded  with  a  weight  uniformly 
distributed  over  its  length,  it  will  sustain  just  twice  the  load 
which  can  be  sustained  at  the  end. 

132.  —  Levers:  Breadth  or  Depth.  —  By  a  proper  trans- 
position of  the  factors  in  (29.),  we  obtain— 

L  (30.) 

Z> 

Hence,  to  ascertain  the  breadth  or  depth  of  a  lever  to  sus- 
tain safely  a  given  weight,  we  have  — 


II2  CONSTRUCTION. 

Rule  XXV. — Multiply  four  times  the  given  weight  by 
the  length  of  the  lever,  and  by  the  factor  of  safety ;  divide 
the  product  by  the  value  of  B  for  the  material  of  the  lever, 
in  Table  III.,  and  the  quotient  will  be  equal  to  the  breadth 
multiplied  by  the  square  of  the  depth.  Now,  if  the  breadth 
be  required,  divide  said  quotient  by  the  square  of  the  depth, 
and  this  quotient  will  be  the  required  breadth  ;  but  if, 
instead  of  the  breadth,  the  depth  be  required,  divide  the 
said  quotient  by  the  breadth  ;  then  the  square  root  of  this 
quotient  will  be  the  required  depth. 

Example. — What  should  be  the  size  of  a  cherry  lever  5 
feet  long  to  sustain  safely  250  pounds  at  its  end?  Proceed- 
ing by  the  rule,  taking  the  factor  of  safety  at  5,  we  have 
4x250x5x5  =  25000.  The  value  of  B  for  cherry,  in  Table 
III.,  is  650 ;  and  25000  divided  by  650  gives  a  quotient  of 
38-46.  Now,  if  the  depth  be  fixed  at  4,  then  38-46  divided 
by  (4x4  =)  16  gives  a  quotient  of  2-4,  the  required  breadth. 
But  if  the  breadth  be  fixed  at  2,  then  38-46  divided  by  2 
gives  a  quotient  of  19-23,  the  square  root  of  which  is  4-38, 
the  required  depth.  Therefore,  the  lever  maybe  2-4x4, 
or  2  x  4-f-  inches. 

133. — Deflection:  Relation  to  Weight. — When  a  load  is 
placed  upon  a  beam  supported  at  each  end,  the  beam  bends 
more  or  less ;  the  distance  that  the  beam  descends  under 
the  operation  of  the  load,  measured  at  the  middle  of  .its 
length,  is  termed  its  deflection.  In  an  investigation  of  the 
laws  of  deflection  it  has  been  demonstrated,  and  experiments 
have  confirmed  it,  that  while  the  elasticity  of  the  material 
remains  uninjured  by  the  pressure,  or  is  injured  in  but  a 
small  degree,  the  amount  of  deflection  is  directly  in  propor- 
tion to  the  weight  producing  it ;  for  example,  if  1000  pounds 
laid  upon  a  beam  is  found  to  cause  it  to  deflect  or  descend  at 
the  middle  a  quarter  of  an  inch,  then  2000  pounds  will  cause 
it  to  deflect  half  an  inch,  3000  pounds  will  deflect  it  three 
fourths  of  an  inch,  and  so  on. 

134. — Deflection  :  Relation  to  Dimensions. — In  Table 
III.  are  recorded  the  results  of  experiments  made  to  test  the 


THE   LAW   OF  DEFLECTION.  113 

resistance  of  the  materials  named  to  deflection.  The  fig- 
ures in  the  third  column  designated  by  the  letter  F  (for  flex- 
ure) show  the  number  of  pounds  required  to  deflect  a  unit 
of  material  one  inch.  This  is  an  extreme  state  of  the  case, 
for  in  most  kinds  of  material  this  amount  of  depression 
would  exceed  the  limits  of  elasticity  ;  and  hence  the  rule 
would  here  fail  to  give  the  correct  relation  as  between  the 
dimensions  and  pressure.  For  the  law  of  deflection  as  above 
stated  (the  deflections  being  in  proportion  to  the  weights) 
is  true  only  while  the  depressions  are  small  in  comparison 
with  the  length.  Nothing  useful  is,  therefore,  derived  from 
this  position  of  the  question,  except  to  give  an  idea  of  the 
nature  of  the  quantity  represented  by  the  constant  F\  it 
being  in  reality  an  index  of  the  stiffness  of  the  kind  of  mate- 
rial used  in  comparing  one  material  with  another.  Whatever 
be  the  dimensions  of  the  beam,  F  will  always  be  the  same 
quantity  for  the  same  material  ;  but  among  various  materials 
/''will  vary  according  to  the  flexibility  or  stiffness  of  each 
particular  material.  For  example,  F  will  be  much  greater 
for  iron  than  for  wood  ;  and  again,  among  the  various  kinds 
of  wood,  it  will  be  larger  for  the  stiff  woods  than  for  those 
that  are  flexible.  The  value  of  F,  therefore,  is  the  weight 
which  would  deflect  the  unit  of  material  one  inch,  upon  the 
supposition  that  the  deflections,  from  zero  to  the  depth  of 
one  inch,  continue  regularly  in  proportion  to  the  increments 
of  weight  producing  the  deflections,  or,  for  each  deflection  — 

F  :   I  :  :  W  :  <?, 
from  which  we  have  — 


in  which  cJ  represents  the  deflection  in  inches  corresponding 
to  Wt  the  weight  producing  it.  This  is  for  the  unit  of  ma- 
terial. For  beams  of  larger  dimensions,  investigations  have 
shown  (Transverse  Strains,  Chapters  XIII.  and  XIV.)  that 
the  power  of  a  beam  to  resist  deflection  by  a  weight  at  mid- 
dle is  in  proportion  to  its  breadth  and  the  cube  of  its  depth, 
and  it  is  inversely  in  proportion  to  the  cube  of  the  length  ; 


1 14  CONSTRUCTION. 

or,  when  the  resistance  of  the  unit  of  material  is  measured, 
as  above,  by  — ,  we  have  the  relation  between  it  and  a 
larger  beam  of — 


Putting  this   ratio  in  a  proportion  with  that  of  the  unit  of 
material,  we  have — 

.     ..; . .    ;  ";i>: ' ::  T  'b-r--  : 

which  gives — 

W  _Fbd* 

6          ~  T    ' 
from  which  we  have — 

W=F-^-.  (31.) 


135. — Deflection  :  Weight  when  at  MidklQc. — In  equation 
(31.)  we  have  a  rule  by  which  to  ascertain  what  weight  is 
required  to  deflect  a  given  beam  to  a  given  depth  of  deflec- 
tion ;  this,  in  words  at  length,  is — 

Rule  XXVI.— Multiply  the  breadth  of  the  beam  by  the 
cube  of  its  depth,  and  by  the  given  deflection,  all  in  inches, 
and  by  the  value  of  .Ffor  the  material  of  the  beam,  in  Table 
III.;  divide  the  product  by  the  cube  of  the  length  in  feet, 
and  the  quotient  will  be  the  required  weight  in  pounds. 

Example. — What  weight  is  required  at  the  middle  of  a 
4x12  inch  Georgia-pine  beam  20  feet  long  to  deflect  it 
three  quarters  of  an  inch  ?  The  value  of  F  for  Georgia 
pine,  in  Table  III.,  is  5900;  therefore,  by  the  rule,  we  have 
4 x  i23  x  0-75  x  5900  =  30585600,  which  divided  by  (20x20 
x  20  =)  8000  gives  a  quotient  of  3823-2,  the  required  weight 
in  pounds. 

136.— Deflection:  Breadth  or  Depth,  Weight  at  middle. 

—By  a  transposition  of  equation  (31.),  we  obtain— 


SIZE   FOR  A   GIVEN   DEFLECTION.  115 

a  rule  by  which  may  be  found  the  breadth  or  depth  of  a 
beam,  with  a  given  load  at  middle  and  with  a  given  deflec- 
tion ;  this,  in  words  at  length,  is — 

Rule  XXVII. — Multiply  the  given  load  by  the  cube  of 
the  length  in  feet,  and  divide  the  product  by  the  product  of 
the  deflection  into  the  value  of  F  for  the  material  of  the 
beam,  in  Table  III. ;  then  the  quotient  will  be  equal  to  the 
breadth  of  the  beam  multiplied  by  the  cube  of  its  depth, 
both  in  inches. 

Now,  to  obtain  the  breadth,  divide  the  said  quotient  by 
the  cube  of  the  depth,  and  this  quotient  will  be  the  required 
breadth.  But  if,  instead  of  the  breadth,  the  depth  be  re- 
quired, then  divide  the  said  quotient  by  the  breadth,  and 
the  cube  root  of  this  quotient  will  be  the  required  depth. 
But  if  neither  breadth  nor  depth  be  previously  fixed,  but  it 
be  required  that  they  bear  a  certain  proportion  to  each 
other ;  such  that  d  :  b  : :  i  :  r,  r  being  a  decimal,  then  b  =  rd, 
and  b  d*  —  r  d*  ;  then,  to  find  the  depth,  divide  the  aforesaid 
quotient  by  the  decimal  r,  and  the  fourth  root  (or  the  square 
root  of  the  square  root)  will  be  the  required  depth,  and  this 
multiplied  by  the  decimal  r  will  give  the  breadth. 

Example. — What  should  be  the  size  of  a  spruce  beam  20 
feet  long  between  bearings,  sustaining  2000  pounds  at  the 
middle,  with  a  deflection  of  one  inch  ?  By  the  rule,  the 
weight  into  the  cube  of  the  length  is  2000  x  8000  =  16000000. 
The  value  of  Ffor  spruce,  in  Table  III.,  is  3500;  this  by  the 
deflection  =  i  gives  3500,  which  used  as  a  divisor  in  divid- 
ing the  above  16000000  gives  a  quotient  of  4571  -43.  Now, 
if  the  breadth  be  required,  the  depth  being  fixed,  say,  at  10, 
then  4571-43  divided  by  (lox  lox  10  =)  1000  gives  4-57,  the 
required  breadth.  The  beam  should  be,  say,  4$  by  10  inches. 
But  if  the  depth  be  required,  the  breadth  being  fixed,  say,  at 
4,  then  4571-43  divided  by  4  gives  1142-86,  the  cube  root 
of  which  is  10-46;  so  in  this  case,  therefore,  the  beam  is 
required  to  be  4  x  io£  inches.  Again,  if  the  breadth  is  to 
bear  a  certain  proportion  to  the  depth,  or  that  the  ratio  be- 
tween them  is  to  be,  say,  0-6  to  i,  then  let  r  =  0-6,  and  then 
457i.43  =  o-6^4,  and  dividing  by  0-6,  we  have  7619-05 
=  d\  This  equals  d*xd*\  therefore  the  square  root  of  7619 


1  16  CONSTRUCTION. 

is  87-29,  and  the  square  root  of  this  is  9-343,  the  required 
depth  in  inches.  Now  9-343x0-6  equals  the  breadth,  or 
9.343x0-6=5-6;  therefore  the  beam  is  required  to  be 
5  -6  x  9-  34  inches,  or,  say,  5f  x  9^  inches. 

137.—  Deflection  :  when  Weight  i§  at  Middle.  —  By  a  trans- 
position of  the  factors  in  (32.),  we  obtain  — 


a  rule  by  which  the  deflection  of  any  given  beam  may  be  as- 
certained, and  which,  in  words  at  length,  is  — 

Rule  XXVIIL—  Multiply  the  given  weight  by  the  cube 
of  the  length  in  feet  ;  divide  the  product  by  the  product  of 
the  breadth  into  the  cube  of  the  depth  in  inches,  multiplied 
by  the  value  of  Fior  the  material  of  the  beam,  in  Table  III., 
and  the  quotient  will  be  the  required  deflection  in  inches! 

Example.  —  To  what  depth  will  1000  pounds  deflect  a 
3x10  inch  white-pine  beam  20  feet  long,  the  weight  being 
at  the  middle  of  the  beam  ?  By  the  rule,  we  have  1000  x  2o3 
=  8000000;  then,  since  the  value  of  F  for  white  pine,  in 
Table  III.,  is  2900,  we  have  3  x  io3  x  2900  =  8700000  ;  using 
this  product  as  a  divisor  and  by  it  dividing  the  former  pro- 
duct, we  obtain  a  quotient  of  0.9195,  the  required  deflection 
in  inches. 

138.  —  Deflection:  Load  .  Uniformly  Distributed.  —  In  two 

beams  of  equal  capacity,  suppose  the  one  loaded  at  the 
middle,  and  the  other  with  its  load  uniformly  distributed 
over  its  length,  and  so  loaded  that  the  deflection  in  one  beam 
shall  equal  that  in  the  other  ;  then  the  weight  at  the  middle 
of  the  former  beam  will  be  equal  to  five  eighths  of  that  on 
the  latter.  This  proportion  between  the  two  has  been  de- 
monstrated by  writers  on  the  strength  of  materials.  (See  p. 
484,  Mechanics  of  Eng.  and  Arch.,  by  Prof.  Mosely,  Am.  eel.  by 
Prof.  Mahan,  1856.)  Hence,  when  £/is  put  to  represent  the 
uniformly  distributed  load,  we  have— 


DEFLECTION   FOR  LOAD    EQUALLY   DISTRIBUTED.        1 17 

or,  when  an  equally  distributed  load  deflects  a  beam  to  a 
certain  depth,  five  eighths  of  that  load,  if  concentrated  at 
the  middle,  would  cause  an  equal  deflection.  This  value  of 
W  may  therefore  be  substituted  for  it  in  equation  (31.),  and 
give— 


from  which  we  obtain  — 

i- 
u=  --  JT  --  >  (34-) 

a  rule  for  a  uniformly  distributed  load. 

139.  —  Deflection:   Weight  when  Uniformly  Distributed. 

—  In  equation  (34.)  we  have  a  rule  by  which  we  may  ascertain 
what  weight  is  required  to  deflect  to  a  given  depth  any 
given  beam.  This,  in  words  at  length,  is  — 

Rule  XXIX.  —  Multiply  1-6  times  the  deflection  by  the 
breadth  of  the  beam,  and  by  the  cube  of  its  depth,  all  in 
inches,  and  by  the  value  of  Ffor  the  material  of  the  beam, 
in  Table  III.  ;  divide  the  product  by  the  cube  of  the  length 
in  feet,  and  the  quotient  will  be  the  required  weight  in 
pounds. 

Example.  —  What  weight,  uniformly  distributed  over  the 
length  of  a  spruce  beam,  will  be  required  to  deflect  it  to  the 
depth  ot  0-5  ot  an  inch,  the  beam  being  3  x  10  inches  and  10 
feet  long?  The  value  of  F  ior  spruce,  in  Table  III.,  is  3500. 
Therefore,  by  the  rule,  we  have  j  -6x0-5  x  3  x  io3  x  3500  = 
8400000,  and  this  divided  by  (10x10x10=)  1000  gives 
8400,  the  required  weight  in  pounds. 

(40,  __  Deflection:  Breadth  or  Depth,  Load  Uniformly 
Distributed.  —  By  transposition  of  the  factors  in  equation 
(54.),  we  obtain  — 


a  rule  for  the  dimensions,  which,  in  words  at  length,  is  — 


Il8  CONSTRUCTION. 

Rule  XXX.  —  Multiply  the  given  weight  by  the  cube  of 
the  length  of  the  beam  ;  divide  the  product  by  i  -6  times  the 
given  deflection  in  inches,  multiplied  by  the  value  of  F  for 
the  material  of  the  beam,  in  Table  III.,  and  the  quotient  will 
equal  the  breadth  into  the  cube  of  the  depth.  Now,  to  ob- 
tain the  breadth,  divide  this  quotient  by  the  cube  of  the  depth, 
and  the  resulting  quotient  will  be  the  required  breadth  in 
inches.  But  if,  instead  of  the  breadth,  the  depth  be  required, 
then  divide  the  aforesaid  quotient  by  the  breadth,  and  the 
cube  root  of  the  resulting  quotient  will  be  the  required  depth 
in  inches.  Again,  if  neither  breadth  nor  depth  be  previously 
determined,  but  to  be  in  proportion  to  each  other  at  a  given 
ratio,  as  r  to  i,  r  being  a  decimal  fixed  at  pleasure,  then  di- 
vide the  aforesaid  quotient  by  the  value  of  r,  and  take  the 
square  root  of  the  quotient;  then  the  square  root  of  this 
square  root  will  be  the  required  depth  in  inches.  The  breadth 
will  equal  the  depth  multiplied  by  the  value  of  the  deci- 
mal r. 

Example.  —  What  should  be  the  size  of  a  locust  beam  10 
feet  long  which  is  to  be  loaded  with  6000  pounds  equally 
distributed  over  the  length,  and  with  which  the  beam  is  to 
be  deflected  £  of  an  inch  ?  The  value  of  F  for  locust,  in 
Table  1  1  1.,  is  5050.  By  the  rule,  we  have  6000  x(io  x  10  x  10  =) 
1000  =  6000000,  which  is  to  be  divided  by  (i  -6xo-  75  x  5050  =) 
6060,  giving  a  quotient  of  990-1.  Now,  if  the  depth  be,  say, 
6  inches,  then  990-  1  divided  by  (6x6x6—)  216  gives  a  quo- 
tient of  4-584,  the  required  breadth  in  inches,  say  4^.  But 
if  the  breadth  be  assumed  at  4  inches,  then  990-  1  divided  by 
4.gives  a  quotient  of  247-  5  2  5,  the  cube  root  of  which  is  6-279, 
the  required  depth  in  inches,  or,  say,  6J.  And,  again,  if  the 
ratio  between  the  breadth  and  depth  be  as  o-  7  to  i,  then  990-  1 
divided  by  0-7  gives  a  quotient  of  1414-43,  the  square  root 
of  which  is  37-609,  of  which  the  square  root  is  6-1326,  the 
required  depth  in  inches,  or,  say,  6J-  ;  and  then  6-1326x0-7  = 
4-293,  the  required  breadth  in  inches;  or,  the  beam  shoujd 
be  4T\  x  6J-  inches. 


14-1.  —  Deflection  :  when  Weight  is  Uniformly  Distributed. 

—By  a  transposition  of  the  factors  of  equation  (35.),  we  ob- 
tain — 


DEFLECTION  OF  LEVERS  AND  BEAMS.         119 


a  result  nearly  the  same  as  that  in  equation  (33.),  which  is  a 
rule  for  deflection  by  a  weight  at  middle,  and  which  by 
slight  modifications  may  be  used  for  deflection  by  an  equally 
distributed  load.  Thus  by— 

Rule  XXXI.—  Proceed  as  directed  in  Rule  XXVIII.  (Art. 
137),  using  the  equally  distributed  weight  instead  of  a  con- 
centrated weight,  and  then  divide  the  result  there  obtained 
for  deflection  by  I  -6  ;  then  the  quotient  will  be  the  required 
deflection  in  inches. 

Example.  —  Taking  the  example  given  under  Rule 
XX  VI  1  1.,  \uArt.  137,  and  assuming  that  the  1000  pounds  load 
with  which  the  beam  is  loaded  be  equally  distributed,  then 
0-9195,  the  result  for  deflection  as  there  found,  divided  by  i  -6, 
as  by  the  above  rule,  gives  0-5747,  the  required  deflection. 
This  result  is  just  five  eighths  of  0-9195,  the  deflection  by  the 
load  at  middle. 

N.B.  —  The  deflection  by  a  uniformly  distributed  load  is 
just  five  eighths  of  that  produced  by  the  same  load  when 
concentrated  at  the  middle  of  the  beam;  therefore,  five 
eighths  of  the  deflection  obtained  by  Rule  XXVIII.  will  be 
the  deflection  of  the  same  beam  when  the  same  weight  is 
uniformly  distributed. 

142.  —  Deflection  of  Levers.  —  The  deflection  of  a  lever  is 
the  same  as  that  of  a  beam  of  the  same  breadth  and  depth, 
but  of  twice  the  length,  and  loaded  at  the  middle  with  a  load 
equal  to  twice  that  which  is  at  the  end  of  the  lever.  There- 
fore, if  P  represents  the  weight  at  the  end  of  a  lever,  and  n 
the  length  of  the  lever  in  feet,  then  2  P=  W  smd  2  n  =  t,  and 
if  these  values  of  Wand  /be  substituted  for  those  in  equa- 
tion (33.),  we  obtain  — 

2  P  x  2  n  3 


which  reduces  to  — 


-.„,,.,  (37-) 

Fbd" 


120  CONSTRUCTION. 

a  result  16  times  that  in  equation  (33.),  which  is  the  deflection 
in  a  beam.  Therefore,  when  a  beam  and  a  lever  equal  in 
sectional  area  and  in  length  be  loaded  by  equal  weights,  the 
one  at  the  middle,  the  other  at  one  end,  the  deflection  of  the 
lever  will  be  16  times  that  of  the  beam.  This  proportion  is 
based  upon  the  condition  that  neither  the  beam  nor  the  lever 
shall  be  deflected  beyond  the  limits  of  elasticity. 

14-3.  —  Deflection  of  a  Lever:  Load  at  End.  —  Equation 
(37.),  in  words  at  length,  is  — 

Ride  XXXII.  —  Multiply  16  times  the  given  weight  by 
the  cube  of  the  length  in  feet  ;  divide  the  product  by  the 
product  of  the  breadth  into  the  cube  of  the  depth  multiplied 
by  the  value  of  .Ffor  the  material  of  the  lever,  in  Table  III., 
and  the  quotient  will  be  the  required  deflection. 

Example.  —  What  would  be  the  deflection  of  a  bar  of 
American  wrought  iron  one  inch  broad,  two  inches  deep, 
loaded  with  150  pounds  at  a  point  5  feet  distant  from  the 
wall  in  which  the  bar  is  imbedded  ?  The  value  of  F  for 
American  wrought  iron,  in  Table  III.,  is  62000.  Therefore, 
by  the  rule,  16  x  150  x  5"  =  300000.  This  divided  by 
(i  x  23  x  62000  =)  496000  gives  0-6048,  the  required  deflec- 
tion —  nearly  f  of  an  inch. 

(4-4-.  —  Deflection  of  a  Lever:  Weight  when  at  End.  —  By 

a  transposition  of  the  factors  in  equation  (37.),  we  obtain  — 


This  result  is  equal  to  one  sixteenth  of  that  shown  in  equa- 
tion (31.),  a  rule  for  the  weight  at  the  middle.  Therefore, 
for— 

Rule  XXXIII.—  Proceed  as  directed  in  Rule  XXVII.; 
divide  the  quotient  there  obtained  by  16,  and  the  resulting 
quotient  will  be  the  required  weight  in  pounds. 

Example.  —  What  weight  is  required  at  the  end  of  a  4  x  12 
inch  Georgia-pine  lever  20  feet  long  to  deflect  it  three 
quarters  of  an  inch?  Proceeding  by  Rule  XXVII.,  we  ob- 
tain a  quotient  of  3823-2;  this  divided  by  16  gives 
say  239,  the  required  weight  in  pounds. 


DEFLECTION    OF   LEVERS   WITH   UNIFORM    LOAD.         121 

145. — Deflection  of  a  Lever  :  Breadth  or  Depth,  Load 
at  End. — A  transposition  of  the  factors  of  equation  (38.) 
gives— 

,   73       i6Pn* 


a  rule  by  which  to  obtain  the  sectional  area  of  the  lever. 
By  comparison  with  equation  (32.)  it  is  seen  that  the  result 
in  (39.)  is  16  times  that  found  by  (32.).  Therefore,  the  dimen- 
sions for  a  lever  loaded  at  the  end  may  be  found  by— 

Rule  XXXIV. — Multiply  by  16  the  first  quotient  found 
by  Rule  XXVII.,  and  then  proceed  as  farther  directed  in 
Rule  XXVII.,  using  the  product  of  16  times  the  quotient, 
instead  of  the  said  quotient. 

Example. — What  should  be  the  size  of  a  spruce  lever  20 
feet  long,  between  weight  and  wall,  to  sustain  2000  pounds 
at  the  end  with  a  deflection  of  I  inch?  Proceeding  by  Rule 
XXVII.,  we  obtain  a  first  quotient  of  4571-43.  By  Rule 
XXXIV.,  4571-43  x  16  —  73144-88.  Now,  if  the  depth  be 
fixed,  say,  at  20,  then  73144-88  divided  by  (20  x  20  x  20  =) 
8000  gives  9- 143,  the  required  breadth.  But  to  obtain  the 
depth,  fixing  the  breadth,  say,  at  9,  we  have  for  73144-88  di- 
vided by  9  =  8127-21,  the  cube  root  of  which  is-  20- 1055,  the 
required  depth.  Again,  if  the  breadth  and  depth  are  to  be 
in  proportion,  say,  as  0-7  to  i-o,  then  73144-88  divided  by 
0-7  gives  104492-7,  the  square  root  of  which  is  323-254,  of 
which  the  square  root  is  17-98,  the  required  depth  in  inches  ; 
and  17-98  x  0-7  =  12-586,  the  required  breadth  in  inches. 
The  lever,  therefore,  should  be,  say,  I2f  x  18  inches. 

146. — Deflection  of  Levers:  Weight  Uniformly  Distrib- 
uted.— A  comparison  of  the  effects  of  loads  upon  levers 
shows  (Transverse  Strains,  Art.  347)  that  the  deflection  by  a 
uniformly  distributed  load  is  equal  to  that  which  would  be 
produced  by  three  eighths  of  that  load  if  suspended  from 
the  end  of  the  lever.  Or,  P  —  f  U.  Substituting  this  value 
of  P,  in  equation  (37.),  gives— 

16  x  |  Un* 

Fb<T*       ' 
which  reduces  to — 


122  CONSTRUCTION. 

a  rule  for  the  deflection  of  levers  loaded  with  an  equally  dis- 
tributed load. 

(47.  _  Deflection  of  Severs  with  Uniformly  Distributed 
Load.  —  The  deflection  shown  in  equation  .(40.)  is  just  six 
times  that  shown  in  equation  (33.).  The  result  by  (33.)  mul- 
tiplied by  6  will  equal  the  result  by  (40.);  therefore,  we 
have  — 

Rule  XXXV.—  Proceed  as  directed  in  Rule  XXVIII.  ; 
the  result  thereby  obtained  multiplied  by  6  will  give  the 
required  deflection. 

Example.  —  To  what  depth  will  500  pounds  deflect  a  3  x  10 
inch  white-pine  lever  10  feet  long,  the  weight  uniformly 
distributed  over  the  lever?  Here,  by  Rule  XXVIII.  ,  we 
obtain  the  result  0-05747  ;  this  multiplied  by  6  gives  0-3448, 
the  required  deflection. 

(4-8.  —  Deflection  of  Levers  :  Weight  when  Uniformly 
Distributed.  —  By  a  transposition  of  factors  in  (40.),  we  ob- 
tain — 


This  is  equal  to  one  sixth  that  of  equation  (31.)  ;  therefore, 
we  have  — 

Rule  XXX  VI.  —  Proceed  as  directed  in  Rule  XXVI.; 
the  quotient  thereby  obtained  divide  by  6,  and  the  quotient 
thus  obtained  will  be  the  required  weight. 

Example.  —  What  weight  will  be  required  to  deflect  a 
4x5  inch  spruce  lever  I  inch,  the  weight  uniformly  dis- 
tributed over  its  length  ?  Proceeding  as  directed  in  Rule 
XXVI.,  the  result  thereby  obtained  is  1750;  this  divided  by 
6  gives  29  if,  the  required  weight  in  pounds. 

14-9.  —  Deflection  of  Severs  :  ISreadth  or  Depth,  Load 
Uniformly  Distributed.  —  A  transposition  of  factors  in  equa- 
tion (41.)  gives  — 


SIMPLICITY   IN    CONSTRUCTION.  123 

This  result  is  just  six  times  that  of  equation  (32.);  we,  there- 
fore, have — 

Rule  XXXVII.— Proceed  as  directed  in  Rule  XXVII.  ; 
multiply  the  first  quotient  thereby  obtained  by  6 ;  then  in 
the  subsequent  directions  use  this  multiplied  quotient  in- 
stead of  the  said  first  quotient,  to  obtain  the  required  breadth 
and  depth. 

Example. — What  should  be  the  size  of  a  spruce  lever  10 
feet  long,  sustaining  2666|  pounds,  uniformly  distributed 
over  its  length,  with  a  deflection  of  I  inch  ?  Proceeding 
by  Rule  XXVII.,  the  first  quotient  obtained  is  761-905; 
this  multiplied  by  6  gives  4571-43,  the  multiplied  quotient 
which  is  to  be  used  in  place  of  the  said  first  quotient.  Now, 
to  obtain  the  breadth,  the  depth  being  fixed,  say,  at  10 ; 
4571  -43  divided  by  (cube  of  10  — )  1000,  the  quotient,  4-57,  is 
the  required  breadth.  But  if  the  breadth  be  fixed,  say,  at 
4,  then,  to  obtain  the  depth,  4571-43  divided  by  4  gives 
1142-86,  the  cube  root  of  which  is  10-46,  the  required  depth. 
Again,  if  the  breadth  and  depth  are  to  be  in  proportion,  say, 
as  0-6  to  i  -o,  then  4571  -43  divided  by  0-6  gives  7619-05,  the 
square  root  of  which  is  87-27,  of  which  the  square  root  is 
9-343,  the  required  depth  in  inches  ;  and  9-343  x  °'6  equals 
5-6,  the  required  breadth  in  inches;  or,  the  lever  may  be, 
say,  5f  x  9§-  inches. 

CONSTRUCTION    IN   GENERAL. 

150.  — Construction:    Object  Clearly  Defined.  —  In    the 

various  parts  of  timber  construction,  known  as  floors,  par- 
titions, roofs,  bridges,  etc.,  each  has  a  specific  object,  and  in 
all  designs  for  such  constructions  this  object  should  be  kept 
clearly  in  view,  the  various  parts  being  so  disposed  as  to 
serve  the  design  with  the  least  quantity  of  material.  The 
simplest  form  is  the  best,  not  only  because  it  is  the  most 
economical,  but  for  many  other  reasons.  The  great  number 
of  joints,  in  a  complex  design,  render  the  construction  liable 
to  derangement  by  multiplied  compressions,  shrinkage,  and, 
in  consequence,  highly  increased  oblique  strains  ;  by  which 
its  stability  and  durability  are  greatly  lessened. 


I24 


CONSTRUCTION. 
FLOORS. 


151. — Floor§  Described. — Floors  are  most  generally  con, 
structed  single;  that  is,  simply  a  series  of  parallel  beams,  each 


FIG.  39. 


spanning  the  width  of  the  building,  as  seen  at  Fig.  39*     Oc- 


FIG.  40. 

casionally  floors  are  constructed  double,  as  at  Fig.  40 ;  and 
sometimes  framed,  as  at  Fig.  41  ;    but   these   methods  are 


RULES   APPLIED    TO   FLOORS. 


I25 


seldom  practised,  inasmuch  as  either  of  these  requires  more 
timber  than  the  single  floor.  Where  lathing  and  plastering 
is  attached  to  the  floor-beams  to  form  a  ceiling  below,  the 
springing  of  the  beams,  by  customary  use,  is  liable  to  crack 
the  plastering.  To  obviate  this  in  good  dwellings,  the  double 
and  framed  floors  have  been  resorted  to,  but  more  in  former 
times  than  now,  as  the  cross-furring  (a  series  of  narrow  strips 
of  board  or  plank  nailed  transversely  to  the  underside  of 


UNIVERSITY 


FIG.    41. 

the  beams  to  receive  the  lathing  for  the  plastering)  serves  a 
like  purpose  very  nearly  as  well. 

152. — Floor-Beams. — The  size  of  floor-beams  can  be  as- 
certained by  the  preceding  rules  for  the  stiffness  of  materials. 
These  rules  give  the  required  dimensions  for  the  various 
kinds  of  material  in  common  use.  The  rules  may  be  some- 
what abridged  for  ordinary  use,  if  some  of  the  quantities 
represented  in  the  formula  be  made  constant  within  certain 
limits.  For  example,  if  the  load  per  foot  superficial  upon 
the  floor  be  fixed,  and  the  deflection,  then  these,  together 
with  the  constant 'represented  by  Fy  may  be  reduced  to  one 


126  CONSTRUCTION. 

constant.  For  dwellings,  the  load  per  foot  may  be  taken  at 
70  pounds,  the  weight  proper  to  be  allowed  for  a  crowd 
of  people  on  their  feet.  (Transverse  Strains,  Art.  114.)  To 
this  add  20  for  the  weight  of  the  material  of  which  the  floor 
is  composed,  and  the  sum,  90,  is  the  value  of  f,  or  the  weight 
per  foot  superficial  for  dwellings.  Then  eft—-.  U  (Art.  130). 
The  rate  of  deflection  allowable  for  this  load  may  be  fixed 
at  0-03  inch  per  foot  of  the  length,  or  d  =.  0-03  /.  Substitut- 
ing these  values  in  equation  (35.),  we  obtain  — 


b  d*=        cfl*  9°c/3          =  1875  */3 

i  -6  Fx  -03  /  ~  ~  i  -6  x  -03  /^  ~         F 

or  — 

.        ,  '       i<r=2*ZLfr.  (43.) 


T  X*7  C 

Putting/  to  represent  —   —  ,  we  have  — 


(44.) 


T  8*7  C 

Now,  by  reducing  —  •—  -,  for  the  six  woods  in  common  use, 
the  value  of  j  for  each  is  found  as  follows: 

Georgia  Pine  .........................  /  =  0-32 

Locust  ......  .........................  j  =  0-37 

White  Oak....  ........................  j  =  0-6 

Spruce  ...............................  j  —  0-54 

White  Pine  ............  '  ...............  j  =  0-65 

Hemlock  ....  ..........................  j  ^0-67 

Equation  (44.")  is  a  rule  for  the  floor-beams  of  dwellings  ; 
it  may  be  used  also  to  obtain  the  dimensions  of  beams  for 
stores  for  all  ordinary  business-  for  it  will  require  from  3  to 
5  times  the  weight  used  in  this  rule,  or  from  200  to  400 
(average  300)  pounds  to  increase  the  deflection  to  the  limit 
of  elasticity  in  beams  of  the  usual  depths  and  lengths.  For 
light  stores,  therefore,  loaded,  say,  to  150  pounds  per  foot, 
the  beams  would  be  safe,  but  the  deflection  would  be  in- 


CONSTANTS   FOR   USE   IN   THE   RULES.  127 

creased  to  0-06  per  foot.  When  so  great  a  deflection  as  this, 
would  not  be  objectionable  to  the  eye,  then  this  rule  (44.) 
will  serve  for  the  beams  of  light  stores.  But  for  first-class 
stores,  taking  the  rate  of  deflection  at  -04  per  foot,  and  * 
fixing  the  weight  per  superficial  foot  at  275  pounds,  includ- 
ing the  weight  of  the  material  of  which  the  floor  is  con- 
structed, and  letting  k  represent  the  constant,  then — 

bd*=kcl\  (45.) 

and  for — 

Georgia  Pine k  =  0-73 

Locust k  =  0-85 

White  Oak k  =  1-38 

Spruce k  —  1-48 

White  Pine k  =  1-23 

Hemlock.  . , k  —  1-53 

153. — Floor-Beam^  for  Dwellings— To  find  the  dimen- 
sions of  floor-beams  for  dwellings,  when  the  rate  of  deflection" 
is  0-03  inch  per  foot,  or  for  ordinary  stores  when  the  load  is 
about  150  pounds  per  foot,  and  the  deflection  caused  by  this 
weight  is  within  the  limits  of  the  elasticity  of  the  material, 
we  have  the  following  rule  : 

Rule  XXXVIIL— Multiply  the  cube  of  the  length  by 
the  distance  apart  between  the  beams  (from  centres),  both  in 
feet,  and  multiply  the  product  by  the  value  of/  (Art.  152) 
for  the  material  of  the  beam,  and  the  product  will  equal  the 
product  of  the  breadth  into  the  cube  of  the  depth.  Now, 
to  find  the  breadth,  divide  this  product  by  the  cube  of  the 
depth  in  inches,  and  the  quotient  will  be  the  breadth  in 
inches.  But  if  the  depth  is  sought,  divide  the  said  product 
by  the  breadth  in  inches,  and  the  cube  root  of  the  quotient 
will  be  the  depth  in  inches ;  or  if  the  breadth  and  depth  are 
to  be  in  proportion  as  r  is  to  unity,  r  representing  any  re- 
quired decimal,  then  divide  the  aforesaid  product  by  the 
value  of  r,  and  extract  the  square  root  of  the  quotient,  and 
the  square  root  of  this  square  root  will  be  the  depth  re- 
quired in  inches,  and  the  depth  multiplied  by  the  value  of  i 
will  be  the  breadth  in  inches. 


128  CONSTRUCTION. 

Example.  —In  a  dwelling  Or  ordinary  stOre,  what  must  be 
the  breadth  of  the  beams,  when  placed  15  inches  from 
centres,  to  support  a  floor  covering  a  span  of  16  feet,  the 
depth  being  1 1  inches,  the  beams  of  white  oak  ?  By  the 
rule,  4096,  the  cube  of  the  length,  by  i|,  the  distance  from 
centres,  and  by  0-6,  the  value  of  j  for  white  oak,  equals 
3072.  This  divided  by  1331,  the  cube  of  the  depth,  equals 
2-31  inches,  or  2T^  inches,  the  required  breadth.  But  if,  in- 
stead of  the  breadth,  the  depth  be  required,  the  breadth 
being  fixed  at  3  inches,  then  the  product,  3072,  as  above,  di- 
vided by  3,  the  breadth,  equals  1024 ;  the  cube  root  of  this 
is  10-08,  or,  say,  10  inches  nearly.  But  if  the  breadth  and 
depth  are  to  be  in  proportion,  say,  as  0-3  to  i-o,  then  the 
aforesaid  product,  3072,  divided  by  0-3,  the  value  of  r, 
equals  10240,  the  square  root  of  which  is  101-2,  and  the 
square  root  of  this  is  10-06,  the  required  depth.  This 
multiplied  by  0-3,  the  value  of  r,  equals  3-02,  the  re- 
quired breadth ;  the  beam  is  therefore  to  be,  say,  3  x  10 
inches. 

154. — Floor-Beams  for  First-Class  Stores. — To  find  the 
breadth  and  depth  of  the  beams  for  a  floor  of  a  first-class 
store  sufficient  to  sustain  250  pounds  per  foot  superficial 
(exclusive  of  the  weight  of  the  material  in  the  floor),  with 
a  deflection  of  0-04  inch  per  foot  of  the  length,  we  have — 

Rule  XXXIX.— The  same  as  XXXVIIL,  with  the  ex- 
ception that  the  value  of  k  (Art.  152)  is  to  be  used  instead 
of  the  value  of  j. 

Example. — The  beams  of  the  floor  of  a  first-class  store 
are  to  be  of  Georgia  pine,  with  a  clear  bearing  between  the 
walls  of  18  feet,  and  placed  14  inches  from  centres:  what 
must  be  the  breadth  when  the  depth  is  1 1  inches  ?  By  the 
rule,  5832,  the  cube  of  the  length,  and  i£,  the  distance  from 
centres,  and  0-73,  the  value  of  k  for  Georgia  pine,  all  multi- 
plied together  equal  4966-92  ;  and  this  product  divided  by 
1331,  the  cube  of  the  depth,  equals  3-732,  the  required 
breadth,  or  3|  inches. 

But  if,  instead  of  the  breadth,  the  depth  be  required : 
what  must  be  the  depth  when  the  breadth  is  3  inches  ? 


DISTANCE  APART  OF  FLOOR-BEAMS.  129 

The  said  product,  4966-92,  divided  by  3,  the  breadth,  equals 
1655-64,  and  the  cube  root  of  this,  11-83,  or>  sav>  I2  inches, 
is  the  depth  required. 

But  if  the  breadth  and  depth  are  to  be  in  a  given  pro- 
portion, say  0-35  to  i-o,  the  4966-92  aforesaid  divided  by 
0-35,  the  value  of  r,  equals  14191,  the  square  root  of 
which  is  119-13,  and  the  square  root  of  this  square  root  is 
10-91,  or,  say,  n  inches,  the  required  depth.  And  10-91 
multiplied  by  0-35,  the  value  of  r,  equals  3-82,  the  required 
breadth,  say  3^  inches. 

I55« —  Floor -Beams:     Distance    from    Centres.  —  It   is 

sometimes  desirable,  when  the  breadth  and  depth  of  the 
beams  are  fixed,  or  when  the  beams  have  been  sawed  and 
are  now  ready  for  use,  to  know  the  distance  from  cen- 
tres at  which  such  beams  should  be  placed  in  order  that  the 
floor  be  sufficiently  stiff.  By  a  transposition  of  the  factors 
in  equation  (44.),  we  obtain — 

bd* 


In  like  manner,  equation  (45.)  produces — 

_bd* 


(470 


These,  in  words  at  length,  are  as  follows : 

Rule  XL.— Multiply  the  cube  of  the  depth  by  the  breadth, 
both  in  inches,  and  divide  the  product  by  the  cube  of  the 
length  in  feet  multiplied  by  the  value  of  /,  for  dwellings 
and  for  ordinary  stores,  or  by  k,  for  first-class  stores,  and 
the  quotient  will  be  the  distance  apart  from  centres  in  feet. 

Example.— K  span  of  17  feet,  in  a  dwelling,  is  to  be  cov- 
ered by  white-pine  beams  3x12  inches:  at  what  distance 
apart  from  centres  should  they  be  placed?  By  the  rule, 
1728,  the  cube  of  the  depth,  multiplied  by  3,  the  breadth, 
equals  5184.  The  cube  of  17  is  4913  ;  this  by  0-65,  the  value 
of  j  for  white  pine,  equals  3193-45-  The  aforesaid  5184 
divided  by  this  3193-45  equals  1-6233  feet,  or,  say,  20  inches. 


CONSTRUCTION. 


156.  —  Framed   Openings   for   Chimney*    and    Stairs. — 

Where  chimneys,  flues,  stairs,  etc.,  occur  to  interrupt  the 
bearing,  the  beams  are  framed  into  a  piece,  b  (Fig.  42),  called 
a  header.  The  beams,  a  a,  into  which  the  header  is  framed 
are  called  trimmers  or  carriage-beams.  These  framed  beams 
require  to  be  made  thicker  than  the  common  beams.  The 
header  must  be  strong  enough  to  sustain  one  half  of  the 
weight  that  is  sustained  upon  the  &w7-beams,  c  c  (the  wall  at 
the  opposite  end  or  another  header  there  sustaining  the  other 
half),  and  the  trimmers  must  each  sustain  one  half  of  the 
weight  sustained  by  the  header  in  addition  to  the  weight  it 
supports  as  a  common  beam.  It  is  usual  in  practice  to  make 


these  framed  beams  one  inch  thicker  than  the  common  beams 
for  dwellings,  and  two  inches  thicker  for  heavy  stores.  This 
practice  in  ordinary  cases  answers  very  well,  but  in  extreme 
cases  these  dimensions  are  not  proper.  Rules  applicable 
generally  must  be  deduced  from  the  conditions  of  the  case— 
the  load  to  be  sustained  and  the  strength  of  the  material. 

157. — Breadth  of  Headers. — The  load  sustained  by.  a 
header  is  equally  distributed,  and  is  equal  to  the  superficial 
area  of  the  floor  supported  by  the  header  multiplied  by  the 
load  on  every  superficial  foot  of  the  floor.  This  is  equal  to 
the  length  of  the  header  multiplied  by  half  the  length  of  the 
tail-beams,  and  by  the  load  per  superficial  foot.  Putting  g 


DIMENSIONS    OF  HEADERS.  Ijl 

for  the  length  of  the  header,  n  for  the  length  of  the  tail- 
beams,  and  /  for  the  load  per  superficial  foot  ;  U,  the  uni- 
formly distributed  load  carried  by  the  header,  will  equal  £ 
f  n  g.  By  substituting  for  £/,  in  equation  (35.),  this  value  of 
it,  we  obtain  — 


The  symbols  g  and  /  here  both  represent  the  same  thing, 
the  length  of  the  header  ;  combining  these,  and  for  #  putting 
its  value  gry  we  obtain— 


. 

3-2  Fr 

To  allow  for  the  weakening  of  the  header  by  the  mor- 
tices for  the  tail-beams  (which  should  be  cut  as  near  the 
middle  of  the  depth  of  the  header  as  practicable),  the  depth 
should  be  taken  at,  say,  one  inch  less  than  the  actual  depth. 
With  this  modification,  we  obtain  — 


If  /be  taken  at  90,  and  r  at  0-03,  we  have,  by  reducing— 

h  _  937-  5  "g*  (4Q.) 

-~' 


which  is  a  rule  for  the  breadth  of  headers  for  dwellings  and 
for  ordinary  stores.  This,  in  words,  is  as  follows  : 

Rule  XLL—  Multiply  937-5  times  the  length  of  the  tail- 
beams  by  the  cube  of  the'length  of  the  header,  both  in  feet. 
The  product  divided  by  the  cube  of  one  less  than  the  depth 
multiplied  by  the  value  of  F,  Table  III.,  will  equal  the 
breadth  of  the  header  in  inches  for  dwellings  or  ordinary 
stores. 

Example.—  K  header  of  white  pine,  for  a  dwelling,  is  10 
feet  long,  and  sustains  tail-beams  20  feet  long  ;  its  depth  is 
\2  inches:  what  must  be  its  breadth?  By  the  rule, 
937.  5x20x10*=  18750000.'  This  divided  by  (12-  i)3x  2900^ 


CONSTRUCTION. 


3859900,  equals  4-858,  say  5  inches,  the  required  breadth. 
F  or  first-class  stores,/  should  be  taken  at  275,  and  r  at  0-04. 
With  these  values  the  constants  in  equation  (48.)  reduce  to 
2I48-4375,  or,  say,  2150.  This  gives— 


a  rule  for  the  breadth  of  a  header  for  first-class  stores.  It 
is  the  same  as  that  for  dwellings,  except  that  the  constant 
2150  is  to  be  used  in  place  of  937-5.  Taking  the  same  ex- 
ample, and  using  the  constant  2150  instead  of  937-5,  we 
obtain  1 1  •  14  as  the  required  breadth  of  the  header  for  a  first- 
class  store.  Modifying  the  question  by  using  Georgia  pine 
instead  of  white  pine,  we  obtain  5  -476  as  the  required  thick- 
ness, say  5^  inches. 

158. — Breadth  of  Carriage-Beams. — A  carriage-beam  or 
trimmer,  in  addition  to  its  load  as  a  common  beam,  carries 
one  half  of  the  load  on  the  header,  which,  as  has  been 
seen  in  the  last  article,  is  equal  to  one  half  of  the  superficial 
area  of  the  floor  supported  by  the  tail-beams  multiplied  by 
the  weight  per  superficial  foot  of  the  load  upon  the  floor ; 
therefore,  when  the  length  of  the  header  in  feet  is  repre- 
sented by  g,  and  the  length  of  the  tail-beams  by  n,  w  equals 

-  x  -  x  /,  equals  £  f  g  n* 

For  a  load  not  at  middle,  we  have  (25.) — 

4  W ' amn 
b  d   =  —BJ— 

*The  load  from  the  header,  instead  of  being  \fg  n,  is,  more  accurately, 
i/«(g  —  c)  :  because  the  surface  of  floor  carried  by  the  header  is  only 
that  which  occurs  between  the  surfaces  carried  by  the  carriage-beams,  each  of 
which  carries  so  much  of  the  floor  as  extends  half  way  to  the  first  tail-beam 

from  it,  or  the  distance  -  ;  therefore,  the  width  of  the  surface  carried  equals 

the  length  of  the  header  less  (  2  x  -  =  W,  or  g  —  c.  When,  however,  it  is  con- 
sidered that  the  carriage-beam  is  liable  to  receive  some  weight  from  a  stairs  or 
other  article  in  the  well-hole,  the  small  additional  load  above  referred  to  is 
not  only  not  objectionable,  but  is  really  quite  necessary  to  be  included  in  the 
calculation. 


THICKNESS   OF   CARRIAGE-BEAMS.  133 

This  is  a  rule  based  upon  resistance  to  rupture.     By  substi- 

7?  / 
tuting  for  a,  the  factor  of  safety,  -p-j-,  its  value  in  terms  of 

resistance  to  flexure  {Transverse  Strains,  (154.)),  we  have  — 

h  j*  —  4  W  B  Im  n  __  4  Wm  n  . 
BIFdr  Fdr 


In  this  expression,  W  is  a  concentrated  weight  at  the  dis- 
tances m  and  n  from  the  two  ends  of  the  beam.  Taking  the 
load  upon  a  carriage-beam  due  to  the  load  from  the  header, 
as  above  found,  and  substituting  it  for  W,  we  obtain  — 


,,*  __  fgmn* 

bd  =  -- 


This  is  the  expression  required  for  the  concentrated  load. 
To  this  is  to  be  added  the  uniformly  distributed  load  upon 
the  carriage-beam  ;  this  is  given  in  equation  (35.).  Substi- 
tuting for  U  of  this  equation  its  value,  fc  /,  gives  — 


,  ,, 

-T67^"       Fr 
Combining  these  two  equations,  we  have  for  the  total  load  — 


r  r 


If,  in  this  equation,  /be  taken  at  90,  and  r  at  0-03,  these 
reduce  to  3000  ;  therefore,  with  this  value  of  -,  we  have  — 


/     N 


This  rule  for  the  breadth  of  carriage-beams  with  one 
header,  for  dwellings  and  for  ordinary  stores,  is  put  in  words 
as  follows  : 


134  CONSTRUCTION. 

Rule  XLII.  —  Multiply  the  length  of  the  framed  opening 
by  its  breadth,  and  by  the  square  of  the  length  of  the  tail- 
beams  ;  to  this  product  add  f  of  the  cube  of  the  length  into 
the  distance  of  the  common  beams  from  centres  —  all  in  feet  ; 
divide  3000  times  the  sum  by  the  cube  of  the  depth  in  inches 
multiplied  by  the  value  of  F  for  the  material  of  the  beam,  in 
Table  III.,  and  the  quotient  will  be  the  breadth  in  inches. 

Example.  —  In  a  tier  of  3  x  10  inch  beams,  placed  14  inches 
from  centres,  what  should  be  the  breadth  of  a  Georgia-pine 
carriage-beam  20  feet  long,  carrying  a  header  12  feet  long, 
.having  tail-beams  1  5  feet  long?  Here  the  framed  opening 
is  5-x  1  2  feet.  Therefore,  according  to  the  rule,  12  x  5  x  15'  = 
13500;  to  which  add(|x2o3x  jf  =)5833i;  the  sum  is  19333^, 
and  this  by  3000=  58000000.  The  value  of  F  for  Georgia 
pine,  in  Table  III.,  is  5900;  the  cube  of  the  depth  is  1000; 
the  product  of  these  two  is  5900000;  therefore,  dividing 
the  above  58000000  by  5900000  gives  a  quotient  of  9.83, 
the  required  breadth  in  inches.  If,  in  equation  (51.),  f  be 

taken  at  275,  and  r  at  0-04,  then  -  becomes  6875,  and  the 
equation  becomes  — 

, 
~~ 


Fd* 

a  rule  for  the  breadth  of  carriage-beams  for  first-c  lass  stores  ; 
the  same  as  that  for  dwellings,  except  that  the  constant  is 
6875  instead  of  3000. 

fl59.  —  Breadth  of  Carriage-Beams  Carrying  Two  Sets 
of  Tail-Beams.—  r-A  rule  for  this  is  the  same  as  that  for  a  car- 
riage-beam carrying  one  set  of  tail-beams,  if  to  it  there  be 
added  the  effect  of  the  second  set  of  tail-beams.  Equation 
(51.)  with  the  additibn  named  becomes  — 


,       , 

(54° 


in  which  n  is  the  length  of  one  set  of  tail-beams,  and  s  the 
length  of  the  other  set  ;  and  m  +  n  =  I. 


CARRIAGE-BEAMS   WITH   TWO    HEADERS.  135 

If  /  be  taken  at  90,  and  r  at  0-03,  these  two  reduce  to 
3000,  and  we  have  — 

_  3000  [,£•;/  (;;?;;  -Kr2)-f  |  r/3] 
Pd*~ 

9 

a  rule  for  the  breadth  of  a  carriage-beam  carrying  two  sets 
of  headers,  for  dwellings  and  for  ordinary  stores.  It  may 
be  stated  in  words  as  follows  : 

Rule  XLIII.  —  Multiply  the  length  of  the  longer  set  of 
tail-beams  by  the  difference  between  this  length  and  the 
length  of  the  carriage-beam,  and  to  the  product  add  the 
square  of  the  length  of  the  shorter  set  of  tail-beams  ;  mul- 
tiply the  sum  by  the  length  of  the  longer  set  of  tail-beams, 
and  by  the  length  of  the  header  ;  to  this  product  add  f  of 
the  product  of  the  cube  of  the  length  of  the  carriage- 
beam  into  the  distance  apart  from  centres  of  the  common 
beams  ;  multiply  this  sum  by  3000  ;  divide  this  product  by 
the  product  of  the  cube  of  the  depth  in  inches  into  the  value 
of  F  ior  the  material  of  the  carriage-beam,  in  Table  III.,  and 
the  quotient  will  be  the  required  breadth. 

Example.  —  In  a  tier  of  3  x  12  inch  beams,  placed  14  inches 
from  centres,  what  should  be  the  breadth  of  a  spruce  car- 
riage-beam 20  feet  long  in  the  clear  of  the  bearings,  carry- 
ing two  sets  of  tail-beams,  one  of  them  9  feet  long,  the 
other  5  feet  ;  the  headers  being  15  feet  long  ?  The  difference 
between  the  longer  set  of  tail-beams  and  the  carriage-beam 
is  (20  —  9  =)  1  1  feet.  Therefore,  by  the  rule,  9  x  1  1  +  5*  = 
124;  then  (124x9x15=)  16740  +  (f  x  203  x  |f  —)  5833^-  = 
22573i;  then  22573^x3000  =  67720000.  Now  the  value  of 
F  for  spruce,  Table  III.,  is  3500;  this  by  I23,  the  cube  of 
the  depth,  equals  6048000;  by  this  dividing  the  aforesaid 
67720000,  we  obtain  a  quotient  of  11-197,  the  required 
breadth  of  the  carriage-beam.  If,  in  equation  (54.),  /  be 
taken  at  275,  and  >  at  0-04,  these  reduce  to  6875,  and  we 
obtain  — 


a  rule  for  the  breadth  of  carriage  -beams  carrying  two  sets 


CONSTRUCTION. 

of  tail-beams,  in  the  floors  of  first-class  stores.  -This  is  like 
the  rule  for  dwellings,  except  that  the  constant  is  6875  in- 
stead of  3000. 

160.  _  Breadth  o£  Carriage  -  Beam  willi  Well-Hole  at 
middle.  —  When  the  framed  opening  between  the  two  sets  of 
tail-beams  occurs  at  the  middle,  or  when  the  lengths  of  the 
two  sets  of  tail-beams  are  equal,  then  equation  (54.)  reduces 
to 


and  if  /be  taken  at  90,  and  r  at  0-03,  these  reduce  to  3000, 
and  we  have— 


a  rule  for  the  breadth  of  a  carriage-beam  carrying  two  sets 
of  tail-beams  of  equal  length,  in  the  floor  of  a  dwelling  or  of 
an  ordinary  store  ;  and  which  in  words  is  as  follows: 

Rule  XLIV.—  Multiply  the  length  of  the  header  by  the 
square  of  the  length  of  the  tail-beams,  and  to  the  product 
add  |  of  the  product  of  the  square  of  the  length  of  the  car- 
riage-beam by  the  distance  apart  from  centres  of  the  com- 
mon beams;  multiply  the  sum  by  3000  times  the  length  of 
the  carriage-beam  ;  divide  the  product  by  the  product  of 
the  cube  of  the  depth  into  the  value  of  F  for  the  material  of 
the  carriage-beam,  in  Table  III.,  and  the  quotient  will  be  the 
required  breadth. 

Example.  —  In  a  tier  of  3x12  inch  beams,  placed  12  inches 
from  centres,  what  must  be  the  thickness  of  a  hemlock  car- 
riage-beam 20  feet  long,  carrying  two  sets  of  tail-beams, 
each  8  feet  long,  with  headers  10  feet  long?  By  the  rule, 
10  x  8"  +  £  x  i  x  20*  —  890  ;  890  x  3000  x  20  =  53400000.  Now, 
the  value  of  F,  in  Table  III.,  for  hemlock  is  2800  ;  this  by  the 
cube  of  the  depth,  1728,  equals  4838400;  by  this  dividing 
the  former  product,  53400000,  and  the  quotient,  11-0367,  is 
the  required  breadth  of  the  carriage-beam. 


CROSS-BRIDGING. 

If,  in  equation  (57.), /be  taken  at  275,  and  rat  0.04,  these 
will  reduce,  to  6875,  and  we  shall  have — 


6875 


(59-) 


a  result  the  same  as  in  equation  (58.),  except  that  the 
constant  is  6875  instead  of  3000.  Equation  (59.)  is  a  rule  for 
the  breadth  of  carriage-beams  carrying  two  sets  of  tail-beams 
of  equal  length,  in  the  floor  of  a  first-class  store.  In  words 
at  length,  it  is  the  same  as  Rule  XLI V.,  except  that  the  con- 
stant 6875  is  to  be  used  in  place  of  3000. 

161.— €ro§§-Bridgiiig,    or   Herring-Bone    Bridging.— The 

diagonal  struts  set  between  floor-beams,  as  in  Fig.  43,  are 
known  as  cross- bridging,  or  herring- 
bone bridging.  By  connecting  the 
beams  thus  at  intervals,  say,  of  from 
5  to  8  feet,  the  stiffness  of  the  floor 
is  greatly  increased.  The  absolute 
strength  of  a  tier  of  beams  to  resist  a 
weight  uniformly  distributed  over 
the  whole  tier  is  augmented  but  lit- 
tle by  cross-bridging  ;  but  the  power 
of  any  one  beam  in  the  tier  to  re- 
sist a  concentrated  load  upon  it,  as  a  heavy  article  of  fur- 
niture or  an  iron  safe,  is  greatly  increased  by  the  cross- 
bridging;  for  this  device,  by  connecting  the  loaded  beam 
with  the  adjacent  beams  on  each  side,  causes  these  beams  to 
assist  in  carrying  the  load.  To  secure  the  full  benefit  of  the 
diagonal  struts,  it  is  very  important  that  the  beams  be  well 
secured  from  separating  laterally,  by  having  strips,  such  as 
cross-furring,  firmly  nailed  to  the  under  edges  of  the  beams. 
The  tie  thus  made,  together  with  that  of  the  floor-plank  on 
the  top  edges,  will  prevent  the  thrust  of  the  struts  from  sep- 
arating the  beams. 

162.— Bridging:  Value  to  Resi§t  Concentrated  L,oad§.— 

A  rule  for  determining  the  additional  load  which  any  one 
beam  connected  by  bridging  will  be  capable  of  sustaining, 
by  the  assistance  derived  from  the  other  beams,  through  the 


FIG.  43. 


138  CONSTRUCTION. 

bridging,  may  be  found  in  Chapter  XVIII.,  Transverse  Strains. 
This  rule  may  be  stated  thus  :  * 


R  =  -(1  +  22  +  3V  42+  etc.)  ;  (60.) 

in  which  R  is  the  increased  resistance,  equal  to  the  addi- 
tional load  which  may  be  put  upon  the  loaded  beam  ;  c  is  the 
distance  from  centres  in  feet  at  which  the  beams  in  the  tier 
are  placed  ;  /is  the  load  in  pounds  per  superficial  foot  upon 
the  floor  ;  /  is  the  length  of  the  beams  in  feet  ;  and  d  is  the 
depth  of  the  beams  in  inches.  The  squares  within  the 
bracket  are  to  be  extended  to  as  many  places  as  there  are 
beams  on  each  side  which  contribute  assistance  through  the 
bridging.  The  rule  given  in  the  work  referred  to,  for  ascer- 
taining the  number  of  spaces  between  the  beams,  is  — 

d  f  ,-    \ 

«  =  7r;  (61.) 

or,  the  depth  of  the  beam  in  inches  divided  by  the  square  of 
the  distance  from  centres,  in  leet,  at  which  the  beams  are 
placed  will  give  the  number  of  spaces  between  the  beams 
which  contribute  on  each  side  in  sustaining  the  concentrated 
load.  The  nearest  whole  number,  minus  unity,  will  equal 
the  required  number  of  beams. 

The  value  of  c  for  beams  in  floors  of  dwellings  is  given  in 
equation  (46.),  and  lor  those  in  first-class  stores  in  equation 
(47.).  By  a  modification  of  equation  (34.),  putting  c  f  I  for 
U,  we  have  — 


and- 


c  = 


=  (63.) 

These    equations  give  general    rules    for    the    value    of    c. 


INCREASED    LOAD   BY   CROSS-BRIDGING.  139 

Now,  the  rule,  in  words  at  length,  for  the  resistance  offered 
by  the  adjoining  beams  to  a  weight  concentrated  upon  one 
of  the  beams  sustained  by  cross-bridging  to  the  others,  is  — 

Rule  XLV. — Divide  the  depth  of  the  beam  in  inches  by 
the  square  of  the  distance  apart  from  centres  in  feet  at 
which  the  floor-beams  are  placed ;  from  the  quotient  deduct 
unity,  and  call  the  whole  number  nearest  to  the  remainder 
the  First  Result.  Take  the  sum  of  the  squares  of  the  con- 
secutive numbers  from  unity  to  as  many  places  as  shall  equal 
the  above  first  result ;  multiply  this  sum  by  5  times  the 
length  in  feet,  by  the  load  per  foot  superficial  upon  the  floor, 
and  by  the  fifth  power  of  the  distance  apart  from  centres  in 
feet  at  which  the  beams  are  placed  ;  divide  the  product  by 
4  times  the  square  of  the  depth  in  inches,  and  the  quotient 
will  be  the  weight  in  pounds  required. 

Example.  —  In  a  tier  of  3  x  12  inch  floor-beams  20  feet 
long,  placed  in  a  dwelling  16  inches  from  centres  and  well 
bridged,  what  load  maybe  uniformly  distributed  upon  one  of 
the  beams,  additional  to  the  load  which  that  beam  is  capable 
of  sustaining  safely  when  unassisted  by  bridging?  Here, 
according  to  the  rule,  12  divided  by  (ij+  ij  =  )  i-J  equals 
6f ;  6J  —  i  —  5|,  the  nearest  whole  number  to  which  is  6,  the 
first  result.  The  sum  of  the  square  of  the  first  6  numbers 
equals  (i  +  22  +  3'  +  4*  +  5°  + 62  =)  I  +4  +  9+  16  +  25  +  36  =  91. 
Therefore,  91  x  5  x  20  x  90  x  (|-)r'  =  345 1266.*  The  square  of 
the  depth  (12  x  12  =  )  H4x4  =  576;  by  this  dividing  the 
above  3451266,  we  have  the  quotient  5991-78,  say  5992 
pounds,  the  required  weight.  This  is  the  additional  load 
which  may  be  placed  upon  the  beam.  At  90  pounds  per 
superficial  foot,  the  common  load  on  each  beam,  we  have 


*  The  value  of  c,  16  inches,  equals  £  feet.  The  fifth  power  of  this,  or  (£)6, 
is  obtained  by  involving  both  numerator  and  denominator  to  the  fifth  power, 
and  dividing  the  fifth  power  of  the  former  by  the  fifth  power  of  the  latter  ;  for 

(i)5  =  i_.     For  the  numerator  we  have  4x4x4x4x4=1024,  and  for  the  de- 

35 
nominator  3x3x3x3x3  =  243.     The  former  divided  by  the  latter  gives  as  a 

quotient  4-214,  the  value  of  (j)5.  The  process  of  involving  a  number  to  a  high 
power,  or  the  reverse  operation  of  extracting  high  roots,  may  be  performed  by 
logarithms  with  great  facility.  (See  Art.  427.) 


I4O  CONSTRUCTION. 

90  x  20  x  |  =  2400  as  the  common  load.  To  this  add  5992, 
the  load  sustained  through  the  bridging  by  the  other  beams, 
and  the  sum,  8392  pounds,  will  be  the  total  load  which  may 
be  safely  sustained,  uniformly  distributed,  upon  one  beam — 
nearly  3^  times  the  common  load. 

163.— Crirclers.— When  the  distance  between  the  walls  of 
a  building  is  greater  than  that  which  would  be  the  limit  for 
the  length  of  ordinary  single  beams,  it  becomes  requisite  to 
introduce  one  or  more  additional  supports.  Where  sup- 
ports are  needed  for  a  floor  and  partitions  are  not  desirable, 
it  is  usual  to  use  a  large  piece  of  timber  called  a  girder,  sus- 
tained by  posts  set  at  intervals  of  from  8  to  1 5  feet ;  or,  when 
posts  are  objectionable,  a  framed  construction  called  a 
framed  girder  (Art.  196) ;  or  an  iron  box  called  a  tubular 
iron  girder  (Art.  182).  When  a  simple  timber  girder  is  used 
it  is  advisable,  if  it  be  large,  to  divide  it  vertically  from  end 
to  end  and  reverse  the  two  pieces,  exposing  the  heart  of  the 
timber  to  the  aii»  in  order  that  it  may  dry  quickly,  and  also 
to  detect  decay  at  the  heart.  When  the  halves  are  bolted 
together,  thin  slips  of  wood  should  be  inserted  between 
them  at  the  several  points  at  which  they  are  bolted,  in  order 
to  leave  sufficient  space  for  the  air  to  circulate  freely  in  the 
space  thus  formed  between  them.  This  tends  to  prevent 
decay,  which  will  be  found  first  at  such  parts  as  are  not 
exactly  tight,  nor  yet  far  enough  apart  to  permit  the  escape 
of  moisture.  When  girders  are  required  for  a  long  bear- 
ing, it  is  usual  to  truss  them ;  that  is,  to  insert  between 
the  halves  two  pieces  of  oak  which  are  inclined  towards  each 
other,  and  which  meet  at  the  centre  of  the  length  of  the 
girder  like  the  rafters  of  a  roof-truss,  though  nearly  if  not 
quite  concealed  within  the  girder.  This  and  many  similar 
methods,  though  extensively  practised,  are  generally  worse 
than  useless ;  since  it  has  been  ascertained  that,  in  nearly  all 
such  cases,  the  operation  has  positively  weakened  the  girder. 

A  girder  may  be  strengthened  by  mechanical  contrivance, 
when  its  depth  is  required  to  be  greater  than  any  one  piece 
of  timber  will  allow.  Fig.  44  shows  a  very  simple  yet  invalu- 
able method  of  doing  this.  The  two  pieces  of  which  the  gir- 


CONSTRUCTION   OF   GIRDERS.  14! 

der  is  composed  are  bolted  or  pinned  together,,  having  keys 
inserted  between  to  prevent  the  pieces  from  sliding.  The 
keys  should  be  of  hard  wood,  well  seasoned.  The  two 
pieces  should  be  about  equal  in  depth,  in  order  that  the 
joint  between  them  may  be  in  the  neutral  line.  (See  Arts. 
120,  121.)  The  thickness  of  the  keys  should  be  about  half 
their  breadth,  and  the  amount  of  their  united  thickness 
should  be  equal  to  a  trifle  over  the  depth  and  one  third  of 
the  depth  of  the  girder.  Instead  of  bolts  orpins,  iron  hoops 
are  sometimes  used  ;  and  when  they  can  be  procured,  they 
are  far  preferable.  In  this  case,  the  girder  is  diminished  at 
the  ends,  and  the  hoops  driven  from  each  end  towards  the 
middle.  A  girder  may  be  spliced  if  timber  of  a  sufficient 
length  cannot  be  obtained ;  though  not  at  or  near  the  mid- 


FIG.  44- 

die,  if  it  can  be  avoided.  (See  Art.  87.)  Girders  should 
rest  from  9  to  12  inches  on  each  wall,  and  a  space  should  be 
left  for  the  air  to  circulate  around  the  ends,  that  the  damp- 
ness may  evaporate. 

164.—  Girders :  IMmeniions.— The  size  of  a  girder,  for 
any  special  case,  may  be  determined  by  equations  (21.),  (22.), 
(25,),  (27.),  and  (28.),  to  resist  rupture  ;  and  to  resist  deflection, 
by  equations  (32.)  and  (35.).  For  girders  in  dwellings,  equa- 
tion (44.)  may  be  used.  In  this  case,  the  value  of  c  is  to  be 
taken  equal  to  the  width  of  floor  supported  by  the  girder, 
which  is  equal  to  the  sum  of  the  distances  half  way  to  the 
wall  or  next  bearing  on  each  side.  When  there  is  but  one 


142  CONSTRUCTION. 

girder  between  the  two  walls,  the  value  of  c  is  equal  to  half 
the  distance  between  the  walls.  The  rule  for  girders  for 
dwellings,  in  words,  is — 

Rule  XLVL— Multiply  the  cube  of  the  length  of  the  gir- 
der by  the  sum  of  the  distances  from  the  girder  half  way  to 
the  next  bearing  on  each  side,  and  by  the  value  of  j  for  the 
material  of  the  girder,  in  Art.  152;  the  product -will  equal 
•the  product  of  the  breadth  of  the  girder  into  the  cube  of  the 
depth.  To  obtain  the  breadth,  divide  this  product  by  the 
cube  of  the  depth  ;  the  quotient  will  be  the  breadth.  To 
obtain  the  depth,  divide  the  said  product  by  the  breadth  ; 
the  cube  root  of  the  quotient  will  be  the  depth.  If  the 
breadth  and  depth  are  to  be  in  a  given  proportion,  say  as 
r  :  i-o,  then  divide  the  aforesaid  quotient  by  the  value  of 
r  ;  take  the  square  root  of  the  quotient;  then  the  square  root 
of  this  square  root  will  be  the  depth,  and  the  depth  multi- 
plied by  the  value  of  r  will  be.  the  breadth. 

Example. — In  the  floor  of  a  dwelling,  what  should  be  the 
size  of  a  Georgia-pine  girder  14  feet  long  between  posts, 
placed  at  10  feet  from  one  wall  and  20  feet  from  the  other? 
The  value  of  c  here  is  V~  +  %°  —  ¥  =  J5-  The  value  of  j 
for  Georgia  pine  (Art.  152)  is  0-32.  By  the  rule,  14' x  15  x 
0-32  =  13171-2.  Now,  to  find  the  breadth  when  the  depth 
is  12  inches  ;  13171  -2  divided  by  the  cube  of  12,  or  by  1728, 
gives  a  quotient  of  7-622,  or  7$,  the  required  breadth. 
Again,,  to  find  the  depth,  when  the  breadth  is  8  inches  : 
13171-2  divided  by  8  gives  1646-4,  the  cube  root  of  which  is 
1 1 -808,  or,  say,  uj  inches,  the  required  depth.  But  if 
neither  breadth  nor  depth  have  been  previously  determined, 
except  as  to  their  proportion,  say  as  0-7  to  i-o,  then 
13171-2  divided  by  0-7  gives  18816,  of  which  the  square 
root  is  137-171,  and  of  this  the  square  root  is  11-712,  or,  say, 
I  if  inches,  the  required  depth.  For  the  breadth,  we  have 
11-712  by  0-7  equals  8-198,  or,  say,  8J,  the  required 
breadth.  Thus  the  girder  is  required  to  be  7$-  x  12,  8  x  iij, 
or  8J  xi  if  inches.  This  example  is  one  in  a  dwelling  or 
ordinary  store  ;  (or  first-class  stores  the  rule  for  girders  is  the 
same  as  the  last,  except  that  the  value  of  k  is  to  be  taken 
instead  ofy,  in  Art.  152. 


FIRE-PROOF   TIMBER   FLOORS.  143 

165.  —  Solid   Timber    Floors.  —  Floors    constructed    with 
rolled-iron  beams  and  brick   arches  are  proof  against  fire 
only  to  a  limited  degree  ;  for  experience  has  shown  that  the 
heat,  in  an  extensive  conflagration,  is  sufficiently  intense  to 
deprive  the  iron  of    its  rigidity,    and  consequently  of   its 
strength.     Singular  as  it  may  seem,  it  is  nevertheless  true 
that  wood,  under  certain  circumstances,  has  a  greater  fire- 
resisting  quality  than  iron.     Floors  of  timber  constructed, 
as  is  usual,  with  the  beams  set  apart,  have  but  little  power 
to  resist  fire,  but  if  the  spaces  between  the  beams  be  filled 
up  solid  with   other  beams,  which  thus  close  the  openings 
against  the  passage  of  the  flames,  and  the  under  surface  be 
coated  with  plastering  mortar  containing  a  large  portion  of 
plaster  of    Paris,    and    finished    smooth,  then    this  wooden 
floor  will  resist  the  action  of  fire  longer  than  a  floor  of  iron 
beams  and  brick  arches.     The  wooden  beams  should  be  se- 
cured to  each  other  by  dowels  or  spikes. 

166.  —  Solid   Timber   Floors  for  Dwellings   and  Assem- 
bly-Rooms. —  From  Transverse  Strains,  Art.  702,  we  have  — 


which  may  be  modified  so  as  to  take  this  form  : 


which  is  a  rule  for  the  depth  or  thickness  of  solid  timber 
floors  for  dwellings,  assembly-rooms,  or  office  buildings,  and 
in  which  y  and  h  are  constants  depending  upon  the  mate- 
rial ;  thus,  for  — 


Georgia  Pine  ................  y  —  4»  and  h  = 

Spruce  ......................  ^  =  2i,  "  //  =  0-365 

White  Pine  ..................  y  =  2j,   "  //  =  0-389 

Hemlock  .....................  y  =  2,      '  7*=o-39 

The  rule  may  be  stated  in  words  thus  : 

Rule  XLVIL—  Multiply   the  length  by  the   value  of 


144  CONSTRUCTION. 

and  by  the  value  of  /i,  as  above  given  ;  to  the  product  add 
82  ;  multiply  the  sum  by  the  cube  of  the  length  ;  divide  this 
product  by  0-576  times  the  value  of  F,  in  Table  III.  ;  then 
the  cube  root  of  the  quotient  will  be  the  required  depth  in 
inches. 

Example.  —  What  depth  is  required  for  a  solid  Georgia- 
pine  floor  to  cover  a  span  of  20  feet  ?  For  Georgia  pine 
F=  5900  ;  y,  as  above  given,  equals  4,  and  h  equals  0-314  ; 
therefore,  by  the  rule  — 


d  -  _  =  6.318; 

0-576x5900  3398'4 

or,  the  depth  required  is,  say,  6-32  or  6^  inches. 

167.  —  Solid  Timber  Floors  for  First-Clas§  Si  ores.  —  The 

equation  given  for  first-class  stores,  in  Transverse  Strains, 
Art.  702,  is  — 

»  -(263 


which  may  be  changed  to  this  form  : 


in  which  y  is  as  before,  and  k  for — 

Georgia  Pine  equals 0-4 

Spruce  equals o •  472 

White  Pine  equals 0-502 

Hemlock  equals o  •  506 

This  rule  may  be  put  in  words  the  same  as  Rule  XLVIL, 
except  as  to  the  constants,  which  require  that  263  be  used  in 
place  of  82,  that  k  be  used  in  place  of  /*,  and  that  0-768  be 
used  in  place  of  0-576.  Table  XXI.  of  Transverse  Strains 
contains  the  results  of  computation  showing  the  depths  of 
solid  timber  floors  for  dwellings  and  assembly-rooms  and 
for  first-class  stores,  in  floors  of  spans  varying  from  8  to  30 
feet,  and  for  the  four  kinds  of  timber  before  named. 


IRON   FLOOR-BEAMS.  145 

I6ff.  —  Rolled  -  Iron  Beam*.  —  The  dimensions  of  iron 
beams,  whether  Avrought  or  cast,  are  to  be  ascertained  by 
the  rules  already  given,  when  the  beams  are  of  rectangular 
form  in  their  cross-section;  these  rules  are  applicable  alike 
to  wood  and  iron  (Art.  93),  and  may  be  used  for  any  mate- 
rial, provided  the  constant  appropriate  to  the  given  mate- 
rial be  used.  But  when  the  form  of  cross- 
section  is  such  as  that  which  is  usual  for 
rolled-iron  beams  (Fig.  45),  the  rules 
need  modifying.  Without'  attempting 
to  explain  these  modifications  (referring 
for  this  to  Transverse  Strains,  Art.  457 
and  following  article),  it  may  be  re- 
marked that  the  elements  of  resistance 
to  flexure  in  a  beam  constitute  what  is 

termed  the  Moment  of  Inertia.     This,  in 

.  FIG.  45. 

a  beam  of  rectangular  cross-section,  is 

equal  to  -J^  of  the  breadth  into  the  cube  of  the  depth  ;  or  — 
I=^bd\  (66.) 

This  would  be  appropriate  to  rolled-iron  beams  if  the 
hollow  on  each  side  were  filled  with  metal,  so  as  to  complete 
the  form  of  cross-section  into  a  rectangle.  The  proper  ex- 
pression for  them  may  be  obtained  by  taking  first  the 
moment  for  the  beam  as  if  it  were  a  solid  rectangle,  and 
from  this  deducting  the  moment  for  the  part  which  on  each 
side  is  wanting,  or  for  the  rectangles  of  the  hollows.  In 
accordance  with  this  view  of  the  case,  we  have  — 


b,d^;  (67.) 

in  which  b  is  the  breadth  of  the  beam  or  width  of  the 
flanges  ;  b,  is  the  breadth  of  the  two  hollows,  or  is  equal  to# 
less  the  thickness  of  the  web  or  stem  ;  d  is  the  depth  includ- 
ing top  and  bottom  flanges  ;  and  dt  is  the  depth  in  the  clear 
between  the  top  and  bottom  flanges. 

Now,  if  equation  (32.)  be  divided  by  12,  we  shall  have  — 


146  CONSTRUCTION. 

and  since  -^  b  d*  represents  the  moment  of  inertia,  we  have- 

(68.) 


12  Fd 

This  gives  the  value  of  /  for  a  beam  of  any  form  in  cross- 
section  loaded  at  the  middle.  By  this  equation  the  values 
of  /  have  been  computed  for  rolled -iron  beams  of  many 
sizes,  and  the  results  recorded  in  Table  XVII.,  Transverse 
Strains.  A  few  of  these  are  included  in  Table  IV.,  as  follows  : 

TABLE  IV. — ROLLED- IRON  BEAMS. 


NAME. 

Depth. 

Weight 
per  yard. 

/  = 

NAME. 

Depth. 

Weight 
per  yard. 

/  = 

Trenton 

qo 

7-8d 

Buffalo 

QO 

109-  1  17 

Paterson   .  . 
Phoenix 

5 
t 

3° 
36 

12-082 

I4'3I7 

Phoenix  
Buffalo  

T?* 

150 
QO 

190-63 
151  -436 

Trenton 

•6 

-1O 

°v  761 

Buffalo 

ioi 

IO^ 

17^  -6-1^ 

Phoenix    .  . 
Trenton    .  . 
Buffalo  

7 

•7 

8 

55 
60 
65 

42-43 
46-012 
64-  526 

Trenton  
iBuffalo  
Paterson.  .  .  . 

1 
loj 

12$ 

I2± 

135 

125 
125 

241-478 
286-OI9 
292-05 

Paterson.  .  . 
Phoenix  ...  . 
Phoenix  .  .  . 

8 
9 
9 

So 
?o 

84 

84-735 
92-207 
107-793 

Paterson.  ... 
iBuffalo  
Trenton  

12* 
12* 

!5fV 

170 
1  80 
150 

398-93^ 
418-945 
528-223 

169.  —  Rolled  -Iron    Beams:    Diiuciitioii*  ;    Weight    at 

middle.  —  If,  in  equation  (68.),  there  be  substituted  for  F  its 
value  for  wrought  iron,  as  in  Table  TIL,  we  shall  have  — 


or — 


"  1  2  x  62000  <y  ' 


Wl* 


744000  rf 


(69.) 


This  is  a  rule  by  which  to  ascertain  the  size  of  a  rolled-iron 
beam  to  sustain  a  given  weight  at  middle  with  a  given  de- 
flection, and,  in  words  at  length,  is  as  follows : 

Rule  XLVII1. — Multiply  the  weight  in  pounds  by  the 
cube  of  the  length  in  feet ;  divide  the  product  by  744000 
times  the  deflection  in  inches,  and  the  quotient  will  be  the 


DEFLECTION   OF   IRON   BEAMS.  147 

moment  of  inertia  of  the  required  beam,  and  may  be  found, 
or  the  next  nearest  number,  in  Table  IV.  in  column  headed 
7.  Opposite  to  the  number  thus  found,  to  the  left,  will  be 
found  the  name,  -depth,  and  weight  per  yard  of  the  required 
beam. 

Example.  —  Which  of  the  beams  of  Table  IV.  would  be 
proper  to  carry  10,000  pounds  at  the  middle  with  a  deflection 
of  one  inch,  the  length  between  bearings  being  20  feet  ? 
Here  we  have,  substituting  for  the  symbols  their  values  — 

W  Y3      _ioooox2o3  _  80000000 
~  744000  tf  ~~~  744000  x  i  ~~     744000  3  '  ' 

or,  the  moment  of  inertia  of  the  required  beam  is  107-527,  the 
nearest  to  which,  in  the  table,  is  107-793,  pertaining  to  the 
Phoenix  9-inch,  84-pound  beam.  This,  then,  is  the  required 
beam. 

170.  —  Rolled-Iron  Beams:  Deflection  when  Weight  is 
at  Middle.  —  By  a  transposition  of  symbols  in  equation  (69.), 
we  have  — 

W  lz 

~-~  (70.) 


or  a  rule  for  the  deflection  of  rolled-iron  beams  when  the 
weight  is  at  the  middle.  This,  in  words,  is— 

Rule  XLIX.—  Multiply  the  weight  in  pounds  by  the 
cube  of  the  length  in  feet  ;  divide  the  product  by  744000 
times  the  value  of  /  for  the  given  beam,  and  the  quotient 
will  be  the  required  deflection  in  inches. 

Example.-JN\Mi\.  will  be  the  deflection  of  a  Phoenix  9- 
inch,  70-pound  beam  20  feet  long,  loaded  at  the  middle  with 
7500  pounds  ?  The  value  of  /  for  this  beam,  in  Table  IV., 
is  92-207;  therefore,  substituting  for  the  symbols  their  val- 
ues, and  proceeding  by  the  rule,  we  have— 


__        ___.  = 
"  744000  /  "  "  744000  x  92  -  207 

or,  the  deflection  will  be,  say,  f  of  an  inch. 


148  CONSTRUCTION. 

171. — Rolled -Iron  Beams  :   Weight  when   at   middle.— 

A  transposition  of  factors  in  equation  (70.)  gives — 

IV—  744QQQ  /  <? 

/3  (71.) 

This  is  a  rule  for  the  weight  at  middle,  and,  in  words,  is — 
Rule  L. — Multiply  744000  times  the  value   of  7  by  the 

deflection  in  inches ;  divide  the  product  by  the  cube  of  the 

length,    and   the  quotient   will  be   the   required  weight  in 

pounds. 

Example. — What  weight  at  the  middle  of  a  Buffalo  g-inch, 

go-pound  beam  will  deflect  it  one  inch,  the  length  between 

bearings  being  20  feet  ?     The  value  of  7  for  this  beam,  in 

Table  IV.,  18109-117;  therefore— 

744000  /<?     744000x109-117x1 

^  203  :  IO147 -WSJ 

or,  the  required  weight  is,  say,  10,148  pounds. 

172. — Rolled -Iron  Beam§  :  Weight  at  any  Point. — The 

equation  for  a  load  at  any  point  is  (Transverse  Strains,  Art. 

485)- 

186000  IS 

—r^r,r  -  (72.) 

in  which  m  and  n  represent  the  two  parts  m  feet  into  which 
the  point  where  the  load  rests  divides  the  length.  This,  in 
words,  is  as  follows  : 

Rule  LI. — Multiply  186000  times  the  value  of  7  by  the 
deflection  in  inches ;  divide  the  product  by  the  product  of 
the  length  into  the  rectangle  formed  by  the  two  parts  into 
which  the  point  where  the  load  rests  divides  the  length  ; 
the  quotient  will  be  the  required  weight  in  pounds. 

Example. — What  weight  is  required,  located  at  10  feet 
from  one  end,  to  deflect  i£  inches  a  Paterson  12^-inch,  125- 
pound  beam  25  feet  long  between  bearings  ?  The  value  of 
7 for  this  beam,  in  Table  IV.,  is  292-05  ;  m  —  10,  and  n  =  I 
-  m  =  25  —  10  ^  15  ;  therefore — 


VARYING  WEIGHTS  ON    IRON   BEAMS.  149 

,,,         I86OOO  Id         I86OOO  X  2Q2-O5  X  I  -s 

-  - 


or,  the  required  weight  is,  say,  21,730  pounds. 

(73.  —  Rolled-Iron  Beams:  Dimension*  •  Weight  at  any 
Point.—  By  transposition  of  factors  in  equation  (72.),  we  ob- 
tain — 

Wlmn 
'"  186000  tf*  (73-) 

This  may  be  expressed  in  words  as  follows  : 

Rule  LI  I.  —  Multiply  the  weight  by  the  length,  and  by  the 
rectangle  of  the  two  parts  into  which  the  point  where  the 
weight  rests  divides  the  length  ;  divide  the  product  by  186000 
times  the  deflection,  and  the  quotient  will  be  the  value  of  7, 
which  (or  its  next  nearest  number)  may  be  found  in  Table 
IV.,  opposite  to  which  will  be  found  the  required  beam. 

Example.  —  What  beam  10  feet  long  will  be  required  to 
carry  5000  pounds  at  3  feet  from  one  end  with  a  deflection 
of  0-4  inch?  Here  we  have  m  equal  3,  and  n  equal  7; 
therefore  — 

Wlmn         5000x10x3x7 

-  .  U  '  -      -       J    A    .    J   I  •? 


1  86000  tf  "       1  86000  x  o  •  4 

The  value  of  /is  14-113,  the  nearest  number  to  which  in 
the  table,  is  14-317,  the  moment  of  inertia  of  the  Phoenix  5- 
inch,  36-pound  beam  ;  this,  therefore,  is  the  .beam  required. 

174.  —  Rolled-Iron  Beams:  Dimensions;  Weight  Uniform- 
ly Distributed.—  Since  f  U  =  W(Art.  138),  equation  (69.)  may 
be  modified  by  the  substitution  of  this  value  of  W,  when  we 
obtain  — 


which  reduces  to  — 


7  = 

744000  $' 


/  =  —        p  (74.) 

1  190400  6 


150  CONSTRUCTION. 

a  rule  for  the  dimensions  of  a  beam  for  a  uniformly  distrib- 
uted load,  which,  in  words,  is  as  follows  : 

Rule  LIII.  —  Multiply  the  uniformly  distributed  load  by 
the  cube  of  the  length  ;  divide  the  product  by  1  190400  times 
the  deflection,  and  the  quotient  will  be  the  value  of  /,  corre- 
sponding to  which,  or  to  its  next  nearest  number  will  be 
found  in  Table  IV/the  required  beam. 

Example.  —  What  beam  10  feet  long  is  required  to  sus- 
tain an  equally  distributed  load  of  14,000  pounds  with  a  de- 
flection of  half  an  inch  ?  For  this  we  have— 

14000  x  io8 


1190400x0-5 

This  is  the  moment  of  inertia  of  the  required  beam  ;  nearly. 
the  same  as  23-761,  in  Table  IV.,  the  value  of  /  fora  Tren- 
ton 6-inch,  Ao-pound  beam,  which  will  serve  as  the  re- 
quired beam. 

175.  —  Rolled-Iron  Beam*  :  Deflection  ;  Weight  Uniformly 

Distributed.  —  A  transposition  of  the  factors  in  equation  (74.) 
gives  — 

*=    UIS 

11904007'  (75-) 

a  rule  for  the  deflection  of  a  uniformly  loaded  beam,  and 
which  may  be  put  in  these  words,  namely  : 

Rule  LIV.  —  Multiply  the  uniformly  distributed  load  by 
the  cube  of  the  length  ;  divide  the  product  by  1190400  times 
the  value  of  /,  Table  IV.,  and  the  quotient  will  be  the  re- 
quired deflection. 

Example.  —  To  what  depth  will  14,000  pounds,  uniformly 
distributed,  deflect  a  Buffalo  lo^-inch,  9O-pound  beam  20 
feet  long?  The  value  of  /for  this  beam,  as  per  the  table,  is 
151-  436  ;  therefore  — 


14000  X  20  3 


—?•=•  0-6213  ; 


1  190400  x  151  -436 
or,  the  required  deflection  is,  say,  •£•  of  an  inch. 


IRON   FLOOR-BEAMS   FOR  DWELLINGS.  151 

!  7  6.—  Rolled  -Iron    Beam§:     Weight    when  Uniformly 

Distributed.  —  Equation  (75.),  by  a  transposition  of  factors, 
gives— 

*9  (76.) 


a  rule  for  the  weight  uniformly  distributed,  and  which  may 
be  worded  thus  : 

Rule  LV.  —  Multiply  1190400  times  the  value  of  /,  Table 
IV.,  by  the  deflection  ;  divide  the  product  by  the  cube  of 
the  length,  and  the  quotient  will  be  the  required  weight. 

Example.  —  What  weight  uniformly  distributed  upon  a 
Buffalo  loj-inch,  105  -pound  beam  25  feet  long  between 
bearings  will  deflect  it  f  of  an  inch  ? 

The  value  of  /  for  this  beam,  as  per  Table  IV.,  is  175-645  ; 
therefore  — 

1190400      17 


or,  the  required  weight  is,  say,  10,036  pounds. 

(77.  —  Rolled-Iron  Beam§:  Floors  of  Dwellings  or  As- 
semfoly-  Rooms.—  From  Transverse  Strains,  Art.  500,  we 
have  — 


a  rule  for  the  distance  from  centres  of  rolled-iron  beams  in 
floors  of  dwellings,  assembly-rooms,  or  offices,  where  the 
spaces  between  the  beams  are  filled  in  with  brick  arches  and 
concrete.  In  the  equation,  c  is  the  distance  apart  from  cen- 
tres in  feet,  and  y  is  the  weight  per  yard  of  the  beam.  This, 
in  words,  is  thus  expressed  : 

Rule  LVL—  Divide  255  times  the  value  of  /  by  the  cube 
of  the  length  ;    from  the  quotient  deduct  one  42oth  part  of 
the  weight  of  the  beam  per  yard,  and  the  remainder  will 
the  required  distance  apart  from  centres. 

Example.—  What  should  be  the  distance  apart  from  cen- 


1  5  2  CONSTRUCTION. 

tres  of  Buffalo  I2j-inch,  125  -pound  beams  25  feet  lo.ng 
between  bearings,  in  the  floor  of  an  assembly-room  ?  For 
these  beams,  in  Table  IV.,  /  equals  286-019,  and  y=  125; 
therefore— 

_  255  x  286-oi9_  125  m 
~^J™         ~  420  ' 

-  -298  =  4-37; 


15625        420 

or,  the  required   distance    from  centres  is,  say,  4  feet  4^ 
inches. 

178.  —  Rolled-Iron  Beams  :  Floor§  of  First-Class  Stores. 

—  From  Transverse  Strains,  Art.  504,  we  have  — 


a  rule  for  the  distance  from  centres  of  rolled-iron  beams  in 
the  floor  of  a  first-class  store  ;  the  spaces  between  the  beams 
being  filled  with  brick  arches  and  concrete.  This  rule  may 
be  put  in  words  as  follows: 

Rule  LVIL—  Divide  148-8  times  the  value  of  7  by  the 
cube  of  the  length  ;  from  the  quotient  deduct  one  96oth 
part  of  the  weight  of  the  beam  per  yard,  and  the  remainder 
will  be  the  distance  apart  of  the  beams  from  centres  in 
feet. 

Example.  —  What  should  be  the  distance  apart  from  cen- 
tres of  Buffalo  I2j-inch,  i8o-pound  beams  20  feet  long 
between  bearings,  in  the  floor  of  a  first-class  store?  For 
these  beams  the  value  of  /,  Table  IV.,  is  418-945,  and  the 
value  of  y  is  180;  therefore— 

c  =  I48'8  x  418-945  _  180          6o  . 

203  960 

or,  the  required  distance  from  centres  is,  say,  7  feet  /| 
inches. 


TIE-RODS   FOR   IRON   BEAMS.  153 

179. — Floor- A  roll  es  :  Oeneral  Considerations.  —  In  fill- 
ing the  spaces  between  the  iron  beams  of  a  floor,  the  arches 
should  be  constructed  with  hard  whole  brick  of  good  shape, 
laid  upon  the  supporting  centre  in  contact  with  each  other, 
and  the  joints  thoroughly  filled  with  cement  grout,  and 
keyed  with  slate.  Made  in  this  manner,  the  arches  need  not 
be  over  four  inches  thick  at  the  crown  for  spans  extending 
to  7  or  8  feet,  and  8  inches  thick  at  the  springing,  where 
they  should  be  started  upon  a  proper  skew-back.  The  rise 
of  the  arch  should  not  be  less  than  i£  inches  for  each  foot 
of  the  span. 

180. — Floor  -  Arches ;  Tie -Rods:  Dwellings.  —  From 
Transverse  Strains ',  Art.  507,  we  have — 


d  =  V 0-0198  c  s,  (79.) 

which  is  a  rule  for  the  diameter  in  inches  of  a  tie-rod  for 
an  arch  in  the  floor  of  a  bank,  office  building,  or  assembly- 
room  ;  in  which  d  is  the  diameter  in  inches  of  the  rod,  s  is 
the  span  of  the  arch,  and  c  is  the  distance  apart  between  the 
rods  (s  and  c  both  in  feet).  This  rule  requires  that  the  arch 
rise  i^  inches  per  foot  of  the  span,  and  that  the  brick-work 
and  the  superimposed  load  each  weigh  70  pounds,  or  to- 
gether 140  pounds.  This  rule,  in  words,  is -as  follows: 

Rule  LVIII. — Multiply  the  span  of  the  arch  by  the  dis- 
tance apart  at  which  the  rods  are  placed,  and  by  the  decimal 
0-0198  ;  the  square  root  of  the  product  will  be  the  diameter 
of  the  required  rod. 

Example. — What  should  be  the  diameter  of  the  wrought- 
iron  ties  of  brick  arches  of  5  feet  span,  in  a  bank  or  hall  of 
assembly,  where  the  ties  are  8  feet  apart  ?  For  this  we 

have — 

d  —  1/0-0198  x^8  x  5  =  1/-792  =  0-89; 

or,  the  diameter  of  the  required  rods  should  be,  say,  -J  of  an 
inch. 

181. — Floor-Arches  ;  Tie-Rods:  First-Class  Stores. — From 
the  same  source  as  in  last  article,  we  have — 

d  =  V o- 04527  c~s,  (80.) 


154 


CONSTRUCTION. 


which  is  a  rule  for  the  size  of  tie-rods  for  the  brick  arches 
of  the  floors  of  first-class  stores,  where  the  arches  have  a 
rise  of  i£  inches  for  each  foot  of  the  span,  and  where  the 
weight  of  the  brick  arch  and  concrete  is  not  over  70  pounds 
per  superficial  foot  of  the  floor,  and  the  loading  does  not 
exceed  250  pounds  per  superficial  foot.  As  the  rule  is  the 
same  as  the  one  in  the  preceding  article,  except  the  deci- 
mal, a  recital  of  the  rule,  in  words,  is  not  here  needed.  To 
obtain  the  required  diameter,  proceed  as  directed  in  Rule 
LVIII.,  using  the  decimal  0-04527  instead  of  the  one  there 
given. 

TUBULAR   IRON   GIRDERS. 

182. — Tubular  Iron  Girders:  De§cripfion. — The  use  of 

wooden  beams  for  floors  is  limited  to  spans  of  about  25  feet. 
When  greater  spans  than  this  are  to  be  covered,  some  expe- 
dient must  be  resorted  to  by  which 
intermediate  bearings  for  the  floor- 
beams  may  be  provided.  Wooden 
girders  may  be  used,  but  these 
need  to  be  supported  by  posts  at 
intervals  of  from  10  to  15  feet, 
unless  the  girders  are  trussed,  or 
made  up  of  top  and  bottom  chords, 
struts,  and  ties.  And  even  this 
is  objectionable,  owing  to  the 
height  such  a  piece  of  framing 
requires,  and  which  encumbers 
the  otherwise  free  space  of  the 
hall.  A  substitute  for  the  framed 
girder  has  been  found  in  the 
tubular  iron  girder,  as  in  Fig.  46,  made  of  rolled  plate 
iron  and  angle  irons,  riveted.  They  require  to  be  stiffened 
by  an  occasional  upright  T  iron  along  eachside,  and  a 
cross-head  at  least  at  each  bearing. 

183. — Tubular  Iron  Girders:  Area  of  Flangc§  ;  Load 
at  Middle. — In  wrought-iron  tubular  girders  it  is  usual  to 
make  the  top  and  bottom  flanges  of  equal  thickness.  From 
Transverse  Strains,  Art.  551,  we  have — 


FIG.  46. 


TUBULAR   IRON   GIRDERS.  1 55 

a  rule  for  the  area  of  the  bottom  flange ;  in  which  a'  equals 
the  area  of  the  flange  in  inches,  W  the  weight  in  pounds  at 
the  middle,  /  the  length  and  d  the  depth  of  the  girder,  both 
in  feet,  and  k  the  saTe  load  in  pounds  per  inch  with  which 
the  metal  may  be  loaded,  and  which  is  usually  taken  at 
9000.  The  rule  may  be  stated  thus : 

Rule  LIX. — Multiply  the  weight  by  the  length  ;  divide 
the  product  by  4  times  the  depth  into  the  value  of  k,  and 
the  quotient  will  be  the  required  area  of  the  bottom  flange. 

Example. — In  a  girder  40  feet  long  and  3  feet  high,  to 
carry  75,000  pounds  at  the  middle,  what  area  of  metal  is 
required  in  the  bottom  flange,  putting  k  at  9000?  For  this 
we  have,  by  the  rule — 

.        W I         75000  x  40 
f,'  — —  j__j_ ~    ,  —  ^7  •77'* 

4  d  k      4  x  3  x  9000 

or,  the  area  required  is  27$  inches.  This  is  the  amount  of 
uncut  metal.  An  allowance  is  required  for  that  which  will 
be  cut  by  rivet-holes.  This  is  usually  an  addition  of  one 
sixth. 

184. Tubular  Iron  Girders  :  Area  of  Flanges;  Load  at 

any  Point.— The  equation  suitable  for  this  (Transverse 
Strains,  Art.  553)  is — 

(82-} 

in  which  m  and  n  are  the  distances  respectively  from  the  lo- 
cation of  the  load  to  the  two  ends  of  the  girder.  The  other 
symbols  are  the  same  as  in  the  last  article.  This  rule  may 
be  thus  stated  : 

Rule  LX.— Multiply  the  weight  by  the  values  of  m  and 
of  n ;  divide  the  product  by  the  product  of  the  depth  into 
the  length  and  into  the  value  of  k,  and  the  quotient  will  be 
the  required  area  of  the  bottom  flange. 

Example.— In  a  girder  50  feet  long  between  bearings  and 


156  CONSTRUCTION. 

3J-  feet  high,  what  area  of  metal  is  required  in  the  bottom 
flange  to  sustain  50,000  pounds  at  20  feet  from  one  end,  when 
k  equals  9000  *  By  the  rule,  we  have  — 

rr  m  n          50000  x  20  x  30 


or,  each  flange  requires    19  inches  of  solid  metal  uncut  for 
rivets. 

185.  —  Tubular  Iron  Girder§  :  Area  of  Flange§  ;  Load 
Uniformly  Distributed.  —  The  equation  appropriate  here  is 
(Transverse  Strains,  Art,  555)  — 


This  is  a  rule  by  which  to  obtain  the  area  of  cross-sec- 
tion of  the  bottom  flange  at  any  point  in  the  length  of  the 
girder,  the  load  uniformly  distributed  ;  m  and  n  being  the 
respective  distances  from  the  point  measured  to  the  two 
ends  of  the  girder,  and  U  representing  the  uniformly  dis- 
tributed load  in  pounds.  This,  in  words,  is  described  as 
follows  : 

Rule  LXI.  —  Divide  the  weight  by  the  product  of  twice 
the  depth  into  the  length  and  into  the  value  of  k  ;  then  the 
quotient  multiplied  by  the  values  of  m  and  of  n  will  be  the 
required  area  of  the  bottom  flange  at  the  point  measured, 
the  distance  of  which  from  the  ends  equals  m  and  n. 

Examplc.~\n  a  girder  50  feet  long  and  3^  feet  high,  to 
carry  a  uniformly  distributed  load  of  120,000  pounds,  what 
area  of  cross-section  is  required  in  the  bottom  flange,  at  the 
middle  and  at  intervals  of  5  feet  thence,  to  each  support  ;  k 
being  taken  at  9000?  Here  we  have,  first— 

.   m  n  1  20000  m  n 

a  =  U  —  -jT-r  =  —  -  =  0-038005  m  n. 

2dkl         2x3^x9000x50 

Now,  when  m  =  n  =  25,  we  have  the  middle  point  ;  then  — 
a'  =  0-038095  m  n  =  0-038095  x  25  x  25  =  23-81  ; 


/Til  w  T  i 

SHEARING   STRAIN. 


j&          "^ 

\vv 

SRJ^r 


or,  the   area   of  the   bottom   flange   at  mid-length  is  23 
inches. 

When  ;;/  =  20,  then  n  =  30,  and — 

a'  =  0-038095  X2ox  30  =  22-86; 

or,  the  required  area,  at  5  feet  either  way  from  the  middle, 
is  22-J  inches. 

When  m  =  15,  then  n  =  35,  and — 

a  =  0-038095  x  15  x  35  =  20-0  ; 

or,  at  10  feet  either  way  from  the  middle,  the  required  area 
is  20  inches. 

When  m=  10,  then  n  =  40,  and — 

a'  —  0-038095  x  10x40  =  15-24  ; 

or,  at  15  feet  either  way  from  the  middle,  the  required  area 
is  15^  inches. 

When  m  =  5,  then  n  —  45,  and — 

a'  =  0-038095  x  5x45  =8-57; 

or,  at  20  feet  each  side  of  the  middle,  the  required  area  is  8f 
inches. 

The  area  of  cross-section  found  in  every  case  is  that  of 
the  uncut  fibres ;  to  this  is  to  be  added  as  much  as  will  be 
cut  by  the  rivets.  This  is  usually  about  one  sixth  of  the  area 
given  by  the  rule.  The  top  flange  is  to  be  made  equal  in 
area  to  the  bottom  flange.  The  flanges  are  unvarying  in 
width  from  end  to  end,  the  variation  of  area  being  obtained 
by  varying  the  thickness  of  the  flanges,  and  this  being  at- 
tained by  building  the  flange  in  lamina,  or  plates ;  but  these 
should  not  be  less  than  a  quarter  of  an  inch  thick.  There 
should  be  added  to  the  length  of  the  girder,  in  the  clear, 
about  one  tenth  of  its  length  for  supports  on  the  walls :  thus, 
a  girder  30  feet  long  requires  3  feet  added  for  supports,  or 
1 8  inches  on  each  wall. 

186.— Tubular  Iron  Girdcr§:   Shearing  Strain.— The  top 

and  bottom  flanges  are  provided  of  sufficient  size  to  resist 


158  CONSTRUCTION. 

the  transverse  strain  ;  the  two  upright  plates,  technically 
termed  the  web,  need,  therefore,  to  be  thick  enough  to  resist 
only  the  shearing  strain.  This,  upon  a  beam  uniformly 
loaded,  is  at  the  middle  theoretically  nothing,  but  from 
thence  it  increases  regularly  towards  each  support,  where  it 
equals  half  the  whole  weight.  For  example,  the  girder  of 
Art.  185,  50  feet  long  between  supports,  carries  120,000 
pounds  uniformly  distributed  over  its  length.  In  this  case 
the  shearing  strain  at  the  wall  at  each  end  is  the  half  of 
120,000  pounds,  or  60,000  pounds;  at  5  feet  from  the  wall  it 
is  -fg  or  -J  less,  or  48,000  pounds  ;  at  10  feet  from  the  wall  it 
is  f  less,  or  36,000  pounds  ;  at  15  feet  it  is  24,000  ;  at  20  feet 
it  is  12,000;  and  at  25  feet  or  the  middle,  it  is  nothing. 

187.—  Tubular  Iron   Girder§:  Thickness  of  Wefo.—  The 

equation  appropriate  for  this  is  — 


in  which  t  is  the  thickness  of  the  web  (equal  to  the  sum  of 
the  thicknesses  of  the  two  side  plates),  d  is  the  height  of  the 
plate  (/  and  d  both  in  inches),  G  is  the  shearing  strain,  and  k' 
is  the  effective  resistance  of  wrought  iron  to  shearing  per 
inch  of  cross-section.  This  may  be  put  in  words  as  follows  : 

Rule  LXII.  —  Divide  the  shearing  strain  by  the  product  of 
the  depth  in  inches  into  the  value  of  /£',  and  the  quotient 
will  be  the  thickness  of  the  web,  or  of  the  two  side  plates 
taken  together. 

Example.  —  What  is  the  required  thickness  of  web  in  a 
girder  50  feet  between  bearings,  side  plates  38  inches  high 
between  top  and  bottom  flanges,  and  to  carry  120,000  pounds, 
uniformly  distributed  ?  Here,  putting  the  shearing  resistance 
of  the  plates  at  7000  pounds  per  inch,  we  have  — 


dk'  '     38x7000         266000* 

The  shearing  strain  at  the  supports,  as  in  last  article,  is 
60000 ;  therefore,  we  have  for  this  point— 


LIGHT   IRON   GIRDERS.  159 

60000 
1  "266000"  ^  °'225' 

When  G  =  48000,  then— 

48000 

:  "^660^0"- °'I8; 
and  when  G  =  36000,  then— 

36000 

/  —  -?— —  0-135. 

266000 

Those  nearer  the  middle  of  the  girder  are  still  less  than 
these  ;  and  these  are  all  below  the  practicable  thickness, 
which  is  half  an  inch  for  the  two  plates.  The  plates  ought 
not  in  practice  ever  to  be  made  less  than  a  quarter  of  an 
inch  thick. 


188. — Tubular  Iron  Girder§,  for  Floors  of 
A§§cmbly-Room§,  and  Office  Buildings.— When  the  floors  of 
these  buildings  are  constructed  with  rolled-iron  beams  and 
brick  arches,  then  the  following  (Art.  568,  Transverse  Strains] 
is  the  appropriate  equation  for  the  area  of  cross-section  of 
the  bottom  flange  of  the  girder : 


in  which  a'  is  in  inches,  and  c,  c  ,  d,  /,  m,  and  n  are  in  feet. 
Also,  a'  is  the  area  required  ;  y  is  the  weight  per  yard  of  the 
rolled-iron  beam  of  the  floor ;  c,  their  distances  from  centres  ; 
c',  the  distance  from  centres  at  which  the  tubular  girders  are 
placed,  or  the  breadth  of  floor  carried  by  one  girder;  d,  the 
depth  of  the  girder;  k,  the  effective  resistance  of  the  metal 
per  inch  in  the  flanges  of  the  girder  ;  and  ;//  and  n  are  the 
distances  respectively  from  the  two  ends  of  the  girder  to  the 
point  at  which  the  area  of  cross-section  of  the  bottom  flange 
is  required.  The  rule  may  be  thus  described  : 

Rule  LXIIL— Divide  the  weight  per  yard  of  the  rolled- 
iron  beams  by  3  times  their  distance  from  centres;  to  the 
quotient  add  140  and  reserve  the  sum  ;  deduct  the  length 
in  feet  from  700,  and  with  the  remainder  as  a  divisor  divide 
700 ;  multiply  the  quotient  by  the  above  reserved  sum,  and 


l6o  CONSTRUCTION. 

by  the  value  of  c' ;  divide  the  product  by  the  product  of 
twice  the  depth  into  the  value  of  k,  and  the  quotient  multi- 
plied by  the  values  of  m  and  of  n  will  be  the  required  area 
of  cross-section  of  the  bottom  flange  at  the  point  in  the 
length  distant  from  the  two  ends  equal  to  m  and  n  respec- 
tively. 

Example. — In  a  floor  of  9-inch,  /o-pound  beams,  4  feet 
from  centres,  what  ought  to  be  the  area  of  the  bottom  flange 
of  a  tubular  girder  40  feet  long  between  bearings,  2  feet  8 
inches  deep,  and  placed  17  feet  from  the  walls  or  from  other 
girders ;  the  area  of  the  flange  to  be  ascertained  at  every  5 
feet  of  the  length  :  the  value  of  k  to  be  put  at  9000  ?  Here 
y  •=.  70,  c  =  4,  cf  =  17,  7=40,  and  d=  2f.  Therefore,  by 
the  rule — 

/  70  \      700  17 

a  =  I  14.0+ )      x o x  m  n  : 

V  3  x  47  700  -  40     2  x  2f  x  9000 

a'  —  145  •  8^-  x  I  -0606  x  0-0003 54rf  x  mn\ 
a'  =  0-05478  m  n. 

The  values  of  m  and  n  are — 

At  the  middle m  =  20  ;  n  =  20 

5  feet  from  middle m—  15;  n  =  25 

10     "        "  "       m  •=•  10 ;  n  =  30 

15    . "        "  "       m=    5  ;  n  =  35 

These  give — 

At  the  middle a'  —  0-05478  x  20  x  20  =  21  -91 

"    5  feet  from  middle a'  =  0-05478  x  15  x  25  =  20-54 

"10     "        "  "      a'  =  0-05478  x  10  x  30  —  16-43 

"15     "        "  "      *'  =  0-05478  x    5x35=    9-59 

These  are  the  areas  of  uncut  fibres  at  the  points  named,  in 
the  lower  flange  ;  the  upper  flange  requires  the  same  sizes. 

(89. — Tubular   Iron    Girders,  for   Floors    of  First-Class 

Stores.— The  equation  proper  for  this  is  (Transverse  Strains, 
Art.  570)— 


CAST   IRON   COMPARED   WITH   WROUGHT.  l6l 


a  rule  the  same  in  form  as  that  of  the  previous  article  ;  hence 
it  needs  no  particular  exemplification. 

Rule  LXIII.  of  last  article  may  be  used  for  this  case, 
simply  by  using  the  constant  320  in  place  of  that  of  140. 

CAST-IRON   GIRDERS. 

190.  —  Cast-iron  Girders:  Inferior.  —  Rolled-  iron  beams 
have  been  so  extensively  introduced  within  a  few  years  as 
to  have  superseded  almost  entirely  the  formerly  much  used 
cast-iron  beam  or  girder.  The  tensile  strength  of  cast  iron 
is  far  inferior  to  that  of  wrought  iron.  This  inferiority  and 
the  contingencies  to  which  the  metal  is  subject  in  casting 
render  it  very  untrustworthy  ;  it  should  not  be  used  where 
rolled-iron  beams  can  be  procured.  A  very  substantial  gir- 
der to  carry  a  brick  wall  is  made  by  placing  two  or  more 
rolled-iron  beams  side  by  side,  and  securing  them  together 
by  bolts  at  mid-height  of  the  web  ;  placing  thimbles  or  sep- 
arators at  each  bolt.  As  there  may  be  cases,  however,  in 
which  cast-iron  girders  will  be  used,  a  few  rules  for  them 
will  here  be  given. 

1  9  1.  _Cast-Iron  Girder:  Load  at  Middle.  —  The  form    of 
cross-section  given  to  this  girder  usually  is  as  shown  in 
Fig.  47. 

In  the  cross-section,  the  bottom  flange 
is  made  to  contain  in  area  four  times  as 
much  as  the  top  flange.  The  strength 
will  be  in  proportion  to  the  area  of  the 
bottom  flange,  and  to  the  height  or 
depth  of  the  girder  at  middle.  Hence, 
to  obtain  the  greater  strength  from  a 
given  amount  of  material,  it  is  requisite 
to  make  the  upright  part,  or  the  web, 
rather  thin  ;  yet,  in  order  to  prevent 
injurious  strains  in  the  casting  while  it 
is  cooling,  the  parts  should  be  nearly 
equal  in  thickness.  The  thickness  of  the  three  parts—  web, 


1  62  CONSTRUCTION. 

top  flange,  and  bottom  flange  —  may  be  made  in  proportion 
as  5,  6,  and  8. 

For  a  weight  at  middle,  the  form  of  the  web  should  be 
that  of  a  triangle  ;  the  top  flange  forming  two  straight  lines 
declining  from  the  centre  each  way  to  the  bottom  flange  at 
the  ends,  like  the  rafters  of  a  roof  to  its  tie-beam.  From 
Transverse  Strains,  Art.  583,  we  have  — 


which  is  a  rule  for  the  area  in  inches  of  the  bottom  flange, 
for  a  load  at  middle  ;  the  area  of  the  top  flange  is  to  be  equal 
to  one  fourth  of  that  of  the  bottom  flange.  To  secure  this, 
make  the  width  of  the  top  flange  equal  to  one  third  of  the 
width  of  the  bottom  flange  ;  the  thickness  of  the  former, 
as  before  directed,  being  made  equal  to  f  or  f  of  the  lat- 
ter. The  weight  W  is  in  pounds  ;  the  length  /  is  in  feet  ; 
and  the  depth  d  is  in  inches.  The  factor  of  safety  a  should 
be  taken  at  not  less  than  3  ;  better  at  4  or  5. 

The  equation  in  words  may  be  as  follows  : 

Rule  LXIV.—  Multiply  the  weight  by  the  length,  and  by 
the  factor  of  safety  ;  divide  the  product  by  4850  times  the 
depth  at  middle,  and  the  quotient  will  be  the  area  in  inches 
of  the  bottom  flange  ;  divide  this  area  by  the  width  of  the 
bottom  flange,  and  the  quotient  will  be  its  thickness.  Of  the 
top  flange  make  its  width  equal  one  third  that  of  the  bot- 
tom flange,  and  its  thickness  equal  to  three  quarters  that  of 
the  latter.  Make  the  thickness  of  the  web  equal  to  f-  that 
of  the  bottom  flange. 

Example.  —  What  should  be  the  dimensions  of  the  cross- 
section  of  a  cast-iron  girder  20  feet  long  between  bearings, 
and  24  inches  high  at  middle,  where  30,000  pounds  is  to  be 
carried  ;  the  factor  of  safety  being  put  at  5  ? 

Here  we  have  W  —  30000  ;  a  —  5  ;  /  —  20  ;  and  d  = 
24  ;  therefore,  by  the  rule— 

30000  x  5  x  20 


THE    BOWSTRING   GIRDER. 


I63 


This  is  the  area  of  the  bottom  flange.  If  the  width  of  this 
flange  be  12  inches,  then  25-773  divided  by  12  gives  2 -15,  or 
2-J-  full,  as  the  thickness.  One  third  of  12  equals  4,  equals 
the  width  of  the  top  flange  ;  and  j-  of  2  •  1 5  equals  i  •  6 1 ,  or  i  f — 
its  thickness.  The  thickness  of  the  web  equals -|  x  2-15  = 
i  -34  or  i-J  inches. 

192 — Cast-iron  Girder:    Load  Uniformly  Distributed. 

The  equation  suitable  to  this  is — 


Ual 


9700  (f 


(88.) 


a  rule  of  like  form  with  that  of  the  last  article ;  therefore, 
Rule  LXIV.  may  be  used  for  this  case,  simply  by  substitut- 
ing 9700  for  4850. 

193.— Cast -Iron  Bowstring  Girder.— An  arched  girder, 
such  as  that  in  Fig.  48,  is  technically  termed  a  "  bowstring 
girder."  The  curved  part  is  a  cast-iron  beam  of  T  form  in 
section,  and  the  horizon- 
tal line  is  a  wrought-iron 
tie-rod  attached  to  the 
ends  of  the  arch.  This 
girder  has  but  little  to 
commend  it,  and  is  by  no 
means  worthy  the  confi- 


dence   placed    in    it     by  FlG-  48' 

builders,  with  many  of  whom  it  is  quite  popular.  The  brick 
arch  usually  turned  over  it  is  adequate  to  sustain  the  entire 
compressive  force  induced  from  the  load  (the  brick  wall 
built  above  it),  and  it  thereby  supersedes  the  necessity  for  the 
iron  arch,  which  is  a  useless  expense.  The  tie-rod  is  the 
only  useful  part  of  the  bowstring  girder,  but  it  is  usually 
made  too  small,  and  not  infrequently  is  seriously  injured  by 
the  needless  strain  to  which  it  is  subjected  when  it  is 
"  shrunk  in"  to  the  sockets  in  the  ends  of  the  arch.  The  bow- 
string girder,  therefore,  should  never  be  used. 

(94.— Substitute  for  tlie  Bowstring  Girder. — As  the  cast- 
iron  arch  of  a  bowstring  girder  serves  only  to  resist  com- 


164 


CONSTRUCTION. 


pression,  its  place  can  as  well  be  filled  by  an  arch  of  brick, 
footed  on  a  pair  of  cast-iron  skew-backs ;  and  these  held 
in  position  by  a  pair  of  wrought-iron  tie-rods,  as  shown  in 
Fig.  49.  This  system  of  construction  is  preferable  to  the 

bowstring  girder,  in  that  the 
tie-rods  are  not  liable  to  injury 
by  "  shrinking  in,"  and  the 
cost  is  less.  From  Transverse 
Strains,  Art.  596,  we  have — 


D  = 


Ul 


9425  d 


(89.) 


FIG.  49. 


an  equation  in  which  D  is  the 
diameter  in  inches  of  each  of 
the  two  tie-rods  of  the  brick 
arch  ;  U  is  the  load  in  pounds 
uniformly  distributed  over  the  arch  ;  /  is  the  span  of  the 
arch  in  feet ;  and  d,  in  inches,  is  its  versed  sine,  or  its  height 
measured  from  the  centre  of  the  tie-rod  to  the  centre  of  the 
thickness  or  height  of  the  arch  at  middle. 

This  equation  may  be  put  in  words  as  follows : 
Rule  LXV. — Multiply  the  weight  by  the  length  ;  divide 
the  product  by  9425  times  the  depth,  and  the  square  root  of 
the  quotient  will  be  the  diameter  of  each  rod. 

Example. — What  should  be  the  diameter  of  each  of  the 
pair  of  tie-rods  required  to  sustain  a  brick  arch  20  feet  span 
from  centres,  with  a  versed  sine  or  height  at  middle  of  30 
inches,  to  carry  a  brick  wall  12  inches  thick  and  30  feet  high, 
weighing  TOO  pounds  per  cubic  foot?  The  load  upon  this 
arch  will  be  for  so  much  of  the  wall  as  will  occur  over  the 
opening,  which  will  be  about  one  foot  less  than  the  span  of 
the  arch,  or  20  —  i  =  19  feet.  Therefore,  the  load  will 
equal  19  x  3ox  i  x  100  =  57,000.  pounds  ;  and  hence,  U  — 
57000,  /  =  20,  d  —  30,  and,  by  the  rule — 


57000  x  20 


__.  --  =  1/4-0318  =  2. 008; 

9425  x  30 

or,  the  diameter  of  each  rod  is  required  to  be  2  inches. 


STRAINS    REPRESENTED   GRAPHICALLY.  165 

FRAMED    GIRDERS. 

195 — Graphic  Representation  of  Strain§ In  the  first 

part  of  this  section,  commencing  at  Art.  71,  the  metfiod  was 
developed  of  ascertaining  the  strains  in  the  various  parts  of 
a  frame  by  the  parallelogram  or  triangle  of  forces.  The 
method,  so  far  as  there  explained,  is  adequate  to  solve  sim- 
ple cases  ;  but  when  more  than  three  pieces  of  a  frame  con- 
verge in  one  point,  the  task  bv  that  method  becomes  difficult. 
This  difficulty,  however,  disappears  when  recourse  is  had  to 
the  method  known  as  that  of  4<  Reciprocal  Figures,  Frames, 
and  Diagrams  of  Forces,"  proposed  by  Professor  I.  Clerk 
Maxwell  in  1867.  This  is  an  extension  of  the  method  by 
the  triangle  of  forces,  and  may  be  illustrated  as  follows  : 


FIG.  50.  FIG.  51. 

Let  the  lines  in  Fig.  50  represent,  in  direction  and 
amount,  four  converging  forces  in  equilibrium  in  any  frame, 
as,  for  example,  the  truss  of  a  roof;  let  the  lines  in  Fig.  51 
be  drawn  parallel  to  those  in  Fig.  50,  in  the  manner  fol- 
lowing, namely :  Let  the  line  A  B  be  drawn  parallel  with  the 
line  of  Fig.  50  which  is  between  the  corresponding  letters 
A  and  B,  and  let  it  be  of  corresponding  length  ;  from  B  draw 
the  line  B  C  parallel  with  the  line  of  Fig.  50  which  is  be- 
tween the  letters  B  and  C,  and  of  corresponding  length  ; 
then  from  C  draw  CD,  and  from  A  draw  A  D,  respectively 
parallel  with  the  lines  of  Fig.  50  designated  by  the  corre- 
sponding letters,  and  extend  them  till  they  intersect  at  D. 
The  lengths  of  these  two  lines,  the  last  two  drawn,  are  de- 
termined by  the  point  D  where  they  intersect;  their  lengths, 
therefore,  need  not  be  previously  known,.  The  lengths  of 
the  lines  in  Fig.  51  are  respectively  in  proportion  to  the 


l66  CONSTRUCTION. 

several  strains  in  Fig.  50,  provided  these  strains  are  in 
equilibrium.  Fig.  5 1  is  termed  a  closed  polygon  of  forces. 
A  system  of  such  polygons,  one  for  each  point,  in  the  frame 
where  forces  converge,  so  constructed  that  no  line  repre- 
senting a  force  shall  be  repeated,  is  termed  a  diagram  of 
forces.  This  diagram  of  forces  is  a  reciprocal  of  the  frame 
from  which  it  is  drawn,  its  lines  and  angles  being  the  same. 
The  facility  of  tracing  the  forces  in  the  diagram  of  forces 
depends  materially  upon  the  system  of  lettering  here  shown, 
and  which  was  proposed  by  Mr.  Bow,  in  his  excellent  work 
on  the  Economics  of  Construction.  In  this  system  each 
line  of  the  frame  is  designated  by  the  two  letters  which  it 
separates  ;  thus  the  line  between  A  and  B  is  called  line  A  B  ; 
that  between  C  and  D  is  called  line  CD ;  and  so  of  others ; 
and  in  the  diagram  the  corresponding  lines  are  called  by  the 
same  letters,  but  here  the  letters  designating  the  line  are,  as 
usual,  at  the  ends  of  the  line.  Any  point  in  a  frame  where 
forces  converge  is  designated  by  the  several  letters  which 
cluster  around  it;  as,  for  example,  in  Fig.  50,  the  point  of 
convergence  there  shown  is  designated  as  point  A  B  C  D. 

This  invaluable  method  of  denning  graphically  the 
strains  in  the  various  pieces  composing  a  frame,  such  as  a 
girder  or  roof-truss,  is  remarkably  simple,  and  is  of  general 
application.  Its  utility  will  now  be  exemplified  in  its  appli- 
cation to  framed  girders,  and  afterwards  to  roof-trusses. 

196.— Framed  4*irder§. — Girders  of  solid  timber  are  use- 
ful for  the  support  of  floors  only  where  posts  are  admissible 
as  supports,  at  intervals  of  from  8  to  15  feet.  For  unob- 
structed long  spans  it  becomes  requisite  to  construct  a  frame 
to  serve  as  a  girder  (Arts.  163,  182).  A  frame  of  this  kind 
requires  two  horizontal  pieces,  a  top  and  a  bottom  chord, 
and  a  system  of  struts  and  suspension-pieces  by  which 
the  top  and  bottom  chords  are  held  in  position,  and  the 
strains  from  the  load  are  transmitted  to  the  bearings  at  the 
ends  of  the  girders.  Various  methods  of  arranging  these 
struts  and  ties  have  been  proposed.  One  of  the  most  simple 
and  effective  is  shown  in  Fig.  52,  forming  a  series  of  isos- 
celes triangles.  The  proportion  between  the  length  and 
height  of  a  girder  is  important  as  an  element  of  economy 


RULES   FOR   FRAMED,  GIRDERS.  167 

both  of  space  and  cost.  When  circumstances  do  not  control 
in  limiting-  the  height,  it  may  be  determined  by  this  equation 
from  Transverse  Strains,  Art.  624 — 

,      (I75+/)/ 

2400     ;  (90.) 

in  which  d  is  the  depth  or  height  between  the  axes  of  the 
top  and  bottom  chords,  and  /  is  the  length  between  the  cen- 
tres of  bearings  at  the  supports  (d  and  /  both  in  feet).  This 
equation  in  words  is  as  follows : 

Rule  LXVI. — To  the  length  add  175  ;  multiply  the  sum 
by  the  length  ;  divide  the  product  by  2400,  and  the  quotient 
will  be  the  required  height  between  the  axes  of  the  top  and 
bottom  chords. 

Example. — What  should  be  the  depth  of  a  girder  which 
is  40  feet  long  between  the  centres  of  action  at  the  supports? 
For  this  the  rule  gives— 

(175+40^40      s 

2400 

or,  the  proper  depth  for  economy  of  material  is  3  feet  and 
7  inches. 

The  number  of  bays,  panels,  or  triangles  into  which  the 
bottom  chord  may  be  divided  is  a  matter  of  some  considera- 
tion. Usually  girders  from— 

20  to    59  feet  long  should  have  5  bays. 

59   «     85  "        "           "          "  6     " 

85    "  107  "        "           "          "  7      " 

107   "  127  "        "           "          "  8     " 

127    "  146  "        "                      "  9 

(97.— Framed  Girder  and  Diagram  of  Force§. — Let  Fig. 
52  represent  a  framed  girder  of  six  bays  of,  say,  n  feet 
each,  or  of  a  total  length  of  66  feet. 

The  lines  shown  are  the  axial  lines,  or  the  imaginary  lines 
passing  through  the  axes  of  the  several  pieces  composing  the 
frame.  The.  six  arrows  indicate  the  six  pressures  into  which 
the  equally  distributed  load  is  supposed  to  be  divided.  Each 
of  these  is  at  the  apex  of  a  triangle,  the  base  of  which  1 
along  the  lower  chord. 


1 68 


CONSTRUCTION. 


The  spaces  between  the  arrows  are  lettered ;  so,  also,  the 
space  between  the  last  arrow  at  either  end  and  the  point  of 
support  has  a  letter,  and  so  has  each  triangle,  and  there  is 
one  for  the  space  beneath  the  lower  chord.  These  letters 
are  to  be  used  in  describing  the  diagram  of  forces,  as  was 
explained  in  Art.  195.  The  diagram  of  forces  (Fig.  53)  for 
this  girder-frame  is  drawn  as  follows,  namely :  Upon  a  verti- 


FIG.  52. 

cal  line  A  ^Vmark  the  points  A,  O,  P,  Q,  R,  S,  and  N,  at  equal 
distances,  to  represent  the  six  equal  vertical  pressures  indi- 
cated by  the  arrows  in  Fig.  52.  The  equal  distances  A  O, 
OP,  etc.,  may  be  made  of  any  convenient  size;  but  it  will 


HGrF 


FIG.  53. 

serve  to  facilitate  the  measurement  of  the  forces  in  the  dia- 
gram if  they  are  made  by  a  scale  of  equal  parts,  and  the 
number  of  parts  given  to  each  division  be  made  equal  to  the 
number  of  tons  of  2000  pounds  each  which  is  contained  in 
the  pressure  indicated  by  each  arrow.  On  this  vertical  line 
the  distance  A  0  represents  the  load  at  the  apex  of  the  tri- 
angle B,  or  the  point  A  OCB  (Art.  195);  the  distance  OP 


GRAPHICAL   DIAGRAMS   OF   FORCES.  169 

represents  the  weight  at  the  second  arrow,  or  at  the  point 
O  PR  D  C,  and  so  of  the  rest.  If  the  weights  upon  the 
points  in  the  upper  chord  had  been  unequal,  then  the  divi- 
sion of  the  vertical  line  A  N  would  have  had  to  be  corre- 
spondingly unequal,  each  division  being  laid  off  by  the  scale, 
to  accord  with  the  weight  represented  by  each.  The  line 
of  loads,  A  N,  being  adjusted,  the  other  lines  are  drawn  from 
it  (Art.  195),  so  as  to  make  a  closed  polygon  for  the  forces 
converging  at  each  point  of  the  frame,  Fig.  52— commenc- 
ing with  the  point  A  B  T,  Fig.  52,  where  there  are  three 
forces,  namely,  the  force  acting  through  the  inclined  strut 
A  B,  the  horizontal  force  in  B  7",  and  the  vertical  reaction 
A  T  at  the  point  of  support.  This  last  is  equal  to  half  the 
entire  load,  or  equal  to  the  pressure  indicated  by  the  three 
arrows,  A  O,  O  P,  and  P  Q,  and  is  represented  in  Fig.  53  by 
A  Q  or  A  T.  From  the  point  Q  draw  a  horizontal  line  Q  B ; 
this  is  parallel  with  the  force  B  T  of  Fig.  52,  in  the  lower 
chord.  From  the  point  A  draw  A  B  parallel  with  the  strut 
A  B  of  Fig.  52.  This  line  intersects  the  line  B  T  in  B  and 
closes  the  polygon  A  B  TA  ;  the  point  B  defines  the  length 
of  the  lines  A  B  and  B  7",  and  these  lines  measured  by  the 
scale  by  which  the  line  of  loads  was  constructed  give  the 
required  pressures  in  the  corresponding  lines,  A  B  and  B  T, 
of  Fig.  52. 

Taking  next,  the  point  ABCO,  where  four  forces  meet, 
of  which  we  already  have  two,  namely,  the  force  in  the 
strut  A  B  and  the  load  A  O — from  the  point  O  draw  the  hori- 
zontal line  O  C ;  this  is  parallel  to  the  horizontal  force  O  C 
of  Fig.  52.  Now  from  B  draw  B  C  parallel  with  the  suspen- 
sion-piece B  C  of  Fig.  52.  This  line  intersects  O  C  in  C,  and 
the  point  C  limits  the  lines  O  C  and  B  C  and  closes  the  poly- 
gon A  B  C  O  A,  the  four  sides  of  which  are  respectively  in 
proportion  to  the  four  forces  converging  at  the  point  A  B  CO 
of  Fig.  52,  and  when  measured  by  the  scale  by  which  the 
line  of  loads  was  constructed  give  the  required  strains  re- 
spectively in  each.  Taking  next  the  point  B  C  D  T,  where 
four  forces  converge,  of  which  we  already  have  two,  B  C 
and  B  T—  from  B  extend  the  horizontal  line  TB  to  D\  from 
C  draw  CD  parallel  with  CD  of  Fig.  52,  and  extend  it  to  in- 
tersect TD  in  D,  and  thus  close  the  polygon  T  B  C  D  T. 


I/O  CONSTRUCTION. 

The  lines  in  a  part  of  this  polygon  coincide — those  from 
B  to  T\  this  is  because  the  two  strains  B  T  and  D  T,  Fig.  52, 
lie  in  the  same  horizontal  line.  Again,  taking  the  point 
OC D  EP,  where  five  forces  meet,  three  of  which,  O  P,  O  C, 
and  CD,  we  already  have — draw  from  D  the  line  D  E  parallel 
with  D  E  of  Fig.  52,  and  from  P  the  line  PE  horizontally  or 
parallel  with  P  E  of  Fig.  52.  These  two  lines  intersect  at  E 
and  close  the  polygon  PO  C  D  E  P,  the  sides  of  which  meas- 
ure the  forces  converging  in  the  point  PO  CD  E,  Fig:  52. 
Next  in  order  is  the  point  D  E  F  T,  Fig.  52,  where  four  forces 
meet,  two  of  which,  T  D  and  D  E,  are  known.  From  E  draw 
EF parallel  with  EF'm  Fig.  52;  and  from  71,  TF  parallel 
with  TFin  Fig.  52  ;  these  two  lines  meet  in  F  and  close  the 
polygon  TD  E  F  T,  the  sides  of  which  measure  the  required 
strains  in  the  lines  converging  at  the  point  DEFT,  Fig.  52. 
Taking  next  the  point  PEFG  Q9Fig.  52,  where  five  forces 
meet,  of  which  we  already  have  three,  QP,  P£,and  E  F— 
Irom  F  draw  a  line  parallel  with  F  G  of  Fig.  52,  and  from  Q 
a  line  parallel  with  Q  G  of  Fig.  52.  These  two  intersect  at  G 
and  complete  the  polygon  QPEFG  Q,  the  lines  of  which 
measure  the  forces  converging  at  PEFG  Q  in  Fig.  52. 

In  this  last  polygon,  a  peculiarity  seems  to  indicate  an 
error:  the  line  FG  has  no  length  ;  it  begins  and  ends  at  the 
same  point ;  or,  rather,  the  polygon  is  complete  without  it. 
This  is  easily  understood  when  it  is  considered  that  the  two 
lines  FG  and  G  H  do  not  contribute  any  strength  towards 
sustaining  the  loads  P  Q  and  QR,  and  in  so  far  as  these 
weights  are  concerned  they  might  be  dispensed  with,  and 
the  space  occupied  by  the  three  triangles  F ,  G,  and  H  left 
free,  and  be  designated  by  only  one  letter  instead  of  three. 
Thus  it  appears  that  there  are  only  four  instead  of  five  forces 
at  the  point  PEFG  Q,  and  that  the  four  are  represented  by 
the  lines  of  the  polygon  QPEFQ. 

The  peculiarity  above  explained  arises  from  considering 
loads  only  on  the  top  chord :  the  analysis  of  the  case  is  cor- 
rect as  worked  from  the  premises  given ;  but  in  practice 
there  is  always  more  or  less  load  on  the  bottom  chord  at  the 
middle,  which  should  be  considered.  This  will  be  included 
in  a  case  proposed  in  the  next  article.  One  half  of  the  dia- 


LOAD    ON   BOTH   CHORDS. 


I/I 


gram  of  forces  is  now  complete.  The  other  half  being  ex- 
actly the  same,  except  that  it  is  in  reversed  order,  need  not 
here  be  drawn. 

198. — Framed    Girders:    Load    on    Both    Chords Let 

Fig.    54    represent  the   axial  lines    of  a  girder  carrying  an 


FIG.  54. 

equally  distributed  load  on  each  chord,  represented  by  the 
arrows  and  balls  shown  in  the  figure.  Let  each  bay  measure 
IO  feet,  or  the  length  ot  the  girder  be  50  feet,  and  its  height 


/\7 


XXX 


M 


FIG.  55- 

be  4J-  feet.     The  diagram  of  forces  (Fig.  55)  for  this  girder 
is  obtained  thus : 

The  plan  of  the  girder,  Fig.  54>  requires  to  be  lettered 
as  shown ;  having  one  letter  within  each  panel  and  outside 
the  frame,  and  one  between  every  two  weights  or  strains. 
Then,  in  Fig.  55,  mark  the  vertical  line  K  V  at  L,  M,  N,  V, 


1 72  CONSTRUCTION. 

and  Pt  dividing  it  by  scale  into  equal  parts,  corresponding 
with  the  weights  on  the  top  chord  represented  by  the  ar- 
rows. For  example,  if  the  load  at  each  arrow  equals  6J 
tons,  make  K  L,  L  M,  M N,  etc.,  each  equal  to  6^  parts  of  the 
scale.  Then  K P  will  equal  the  total  load  on  the  top  flange. 
Make  the  distance  P  V  equal  to  the  sum  of  the  loads  on  the 
bottom  chord.  Then  K  V equals  the  total  load  on  the  gir- 
der. Bisect  K  Fin  U\  then  K  U or  U  Fequals  half  the  total 
load  ;  consequently,  equals  the  reaction  of  the  bearing  at  K 
or  P  of  Fig.  54. 

Now,  to  obtain  the  polygon  of  forces  converging  at 
K  A  U,  Fig.  54,  we  have  one  of  these  forces,  K  U,  or  the  re- 
action of  the  bearing  at  KA  U,  equal  to  K  U,  Fig.  55.  From 
f/draw  UA  parallel  with  U  A  of  Fig.  54,  and  from  1C  draw 
KA  parallel  with  the  strut  KA,  Fig.  54,  and  intersecting  the 
line  UA  at  A,  a  point  which  marks  the  limit  of  K  A  and  UA, 
and  closes  the  polygon  K  A  UK,  the  sides  of  which  are  in 
proportion  respectively  to  the  three  strains  which  converge 
at  the  point  A  UK,  Fig.  54.  For  example,  since  the  line  K  U 
by  scale  measures  the  vertical  reaction,  K  U,  of  the  bearing 
at  A  UK,  Fig.  54,  therefore  the  line  K  A  of  the  diagram  of 
forces  by  the  same  scale  measures  the  strain  in  the  strut  KA, 
Fig.  54,  and  the  line  A  U  of  the  diagram  by  the  same  scale 
measures  the  strain  in  the  bottom  chord  at  A  U,  Fig.  54.  For 
the  strains  converging  at  K  A  B  L,  Fig.  54,  of  which  two, 
KA  and  K  L,  are  already  known,  we  draw  from  A  the  line 
A  B  parallel  with  the  line  A  B,  Fig.  54,  and  from  L  draw  L  B 
parallel  with  L  B,  Fig.  54,  meeting  A  B  at  B,  a  point  which 
limits  the  two  lines  and  closes  the  polygon  K  A  B  L  K,  the 
lines  of  which  are  in  proportion  respectively  to  the  strains 
converging  at  the  point  KA  B  L,  Fig.  54,  as  before  explained. 
Of  the  five  strains  converging  at  U  A  B  C  T,  we  already  have 
three — T  U,  UA,and  AB*  to  obtain  the  other  two,  make 
UQ  equal  to  PV,  equal  to  the  total  load  upon  the  lower 
flange  ;  divide  U  Q  into  four  equal  parts,  QR,  RS,  S  T,  and 
T  U,  corresponding  with  the  four  weights  on  the  lower 
chord,  and  represented  by  the  four  balls,  Fig.  54.  Now, 
from  T,  the  point  marking  the  first  of  these  divisions,  draw 
TC  parallel  with  T  C,  Fig.  54,  and  from  B  draw  B  C  paral- 


VARIOUS    STRAINS   IN   FRAMED   GIRDERS.  1/3 

lei  with  the  strut  EC,  Fig.  54,  meeting  TC  in  C,  a  point 
which  limits  the  lines  B  C  and  TC  and  closes  the  polygon 
T  U  A  B  C  T,  the  sides  of  which  are  in  proportion  respectively 
to  the  strains  converging  in  the  point  T  U  A  B  C  T,  Fig.  54. 
Of  the  five  forces  converging  at  MLB  CD,  we  already  have 
three— ML,  LB,  and  B  C\  to  obtain  the  other  two,  from  M 
draw  M D  parallel  with  M D,  Fig.  54,  and  from  C  draw  CD 
parallel  with  CD,  Fig.  54,  meeting  MD  at  D,  a  point  limit- 
ing the  lines  M  D  and  CD  and  closing  the  polygon 
MLBCDM,i\\Q  sides  of  which  are  in  proportion  to  the 
strains  converging  at  the  point  MLB  CD,  Fig.  54.  Of  the 
five  forces  converging  at  the  point  5  TCD E,  three — S  T, 
T  C,  and  CD — are  known;  to  obtain  the  other  two,  from  5 
draw  SE  parallel  with  SE,  Fig.  54,  and  from  D  draw  D  E, 
parallel  with  the  strut  D  E,  Fig.  54,  meeting  the  line  SE  in 
Et  a  point  limiting  the  two  lines  S  E  and  D  E  and  closing  the 
polygon  5  TCD  E  S,  the  sides  of  which  are  in  proportion  to 
the  strains  converging  at  5  TCDE,  Fig.  54.  One  half  of  the 
strains  in  Fig.  54  are  now  shown  in  its  diagram  of  forces,  Fig. 
55  ;  and  since  the  two  halves  of  the  girder  are  symmetrical, 
the  forces  in  one  half  corresponding  to  those  in  the  other, 
hence  the  lines  of  the  diagram  for  one  half  of  the  forces 
may  be  used  for  the  corresponding  forces  of  the  other  half. 

199. — Framed    Girders  :     Dimensions    of  Parl§.  —  The 

parts  of  a  framed  girder  are  the  two  horizontal  chords  (top 
and  bottom)  and  the  diagonals— the  struts  and  ties.  The  top 
chord  is  in  a  state  of  compression,  while  the  bottom  chord 
experiences  a  tensile  strain.  Those  of  the  diagonal  pieces 
which  have  a  direction  from  the  top  to  the  bottom  chord, 
and  from  the  middle  towards  one  of  the  bearings  of  the 
girder,  as  KA,  B  C,  or  D  E,  Fig.  54,  are  struts,  and  are  sub- 
jected to  compression.  The  diagonal  pieces  which  have  a 
direction  from  the  bottom  to  the  top  chord,  and  from  the 
middle  towards  one  of  the  supports,  as  A  B  or  CD,  Fig.  54. 
are  ties,  and  are  subjected  to  extension,  (Art.  83).  The 
amount  of  strain  in  each  piece  in  a  framed  girder  having 
been  ascertained  in  a  diagram  of  forces,  as  shown  in  Arts. 
197  and  198,  the  dimensions  of  each  piece  may  be  obtained 


174  CONST  RUCTION. 

by  rules  already  given.  The  dimensions  of  the  pieces  in  a 
state  of  compression  are  to  be  ascertained  by  the  rules  for 
posts  in  Arts.  107  to  114,  and  those  in  a  state  of  tension  by 
A:mts.  117  to  119  (see  Arts.  226  to  229).  Care  is  required,  in 
obtaining  the  size  of  the  lower  chord,  to  allow  for  the  joints 
which  necessarily  occur  in  long  ties,  for  the  reason  that  tim- 
ber is  not  readily  obtained  sufficiently  long  without  splicing. 
Usually,  in  cases  where  the  length  of  the  girder  is  too  great 
to  obtain  a  bottom  chord  in  one  piece,  the  chord  is  made  up 
of  vertical  lamina,  and  in  as  long  lengths  as  practicable,  and 
secured  with  bolts.  A  chord  thus  made  will  usually  require 
about  twice  the  material  ;  or,  its  sectional  area  of  cross-sec- 
tion will  require  to  be  twice  the  size  of  a  chord  which  is  in 
one  whole  piece ;  and  in  this  chord  it  is  usual  to  put  the  fac- 
tor of  safety  at  from  8  to  10. 

The  diagonal  ties  are  usually  made  of  wrought  iron,  and 
it  is  well  to  secure  the  struts,  especially  the  end  ones,  with 
iron  stirrups  and  bolts.  And,  to  prevent  the  evil  effects  of 
shrinkage,  it  is  well  to  provide  iron  bearings  extending 
through  the  depth  of  each  chord,  so  shaped  that  the  struts 
and  rods  may  have  their  bearings  upon  it,  instead  of  upon 
the  wood. 

PARTITIONS. 

200 — Partitions — Such  partitions  as  are  required  for 
the  divisions  in  ordinary  houses  are  usually  formed  by  tim- 
ber of  small  size,  termed  studs  or  joists.  These  are  placed 
upright  at  12  or  16  inches  from  centres,  and  Avell  nailed. 
Upon  these  studs  lath  are  nailed,  and  these  are  covered 
with  plastering.  The  strength  of  the  plastering  depends  in 
a  great  measure  upon  the  clinch  iormed  by  the  mortar  which 
has  been  pressed  through  between  the  lath.  That  this 
clinch  may  be  interfered  with  in  the  least  possible  degree,  it 
is  proper  that  the  edges  of  the  partition-joists  which  are 
presented  to  receive  the  lath  should  be  as  narrow  as  prac- 
ticable ;  those  which  are  necessarily  large  should  be  reduced 
by  chamfering  the  corners.  The  derangements  in  floors, 
plastering,  and  doors  which  too  frequently  disfigure  the 
interior  of  pretentious  houses  with  gaping  cracks  in  the 


FRAMED    PARTITIONS. 


175 


plastering  and  in  the  door-casings  are  due  in  nearly  all  cases 
to  defective  partitions,  and  to  the  shrinkage  of  floor-timbers. 
A  plastered  partition  is  too  heavy  to  be  trusted  upon  an  ordi- 
nary tier  of  beams,  unless  so  braced  as  to  prevent  its  weight 
from  pressing  upon  the  beams.  This  precaution  becomes  es- 
pecially important  when,  in  addition  to  its  own  weight,  the 
partition  serves  as  a  girder  to  carry  the  weight  of  the  floor- 
beams  next  above  it.  In  order  to  reduce  to  the  smallest 
practicable  degree  the  derangements  named,  it  is  important 
that  the  studs  in  a  partition  should  be  trussed  or  braced  so 
as  to  throw  the  weight  upon  firmly  sustained  points  in  the 
construction  beneath,  and  that  the  timber  in  both  partitions 
and  floors  should  be  well  seasoned  and  carefully  framed. 
To  avoid  the  settlement  due  to  the  shrinkage  of  a  tier  of 
beams,  it  is  important,  in  a  partition  standing  over  one  in  the 
story  below  or  over  a  girder,  that  the  studs  pass  between 
the  beams  to  the  plate  of  the  lower  partition,  or  to  the 
girder  ;  and,  to  be  able  to  do  this,  it  is  also  important  to  ar- 
range the  partitions  of  the  several  stories  vertically  over 
each  other.  All  principal  partitions  should  be  of  brick, 
especially  such  as  are  required  to  assist  in  sustaining  the 
floors  of  the  building. 


FIG.  56. 

201.— Examples  of  Partition &.—Fig.  56  represents  a  par- 
tition having  a  door  in  the  middle.  Its  construction  is  simple 
but  effective.  Fig.  57  shows  the  manner  of  constructing  a 


I76 


CONSTRUCTION. 


partition  having-  doors  near  the  ends.  The  truss  is  formed 
above  the  door-heads,  and  the  lower  parts  are  suspended 
from  it.  The  posts  a  and  b  are  halved,  and  nailed  to  the 
tie  c  d  and  the  sill  ef.  The  braces  in  a  trussed  partition 


r 


a  b 

FIG    57- 


should  be  placed  so  as  to  form,  as  near  as  possible,  an  angle 
of  40  degrees  with  the  horizon.  The  braces  in  a  partition 
should  be  so  placed  as  to  discharge  the  weight  upon  the 


FIG.  58. 

points  of  support.    All  oblique   pieces   that  fail  to  do  this 
should  be  omitted. 

When  the  principal  timbers  of  a  partition  require  to  be 
large  for  the  purpose  of  greater  strength,  it  is  a  good  plan 


WEIGHT  UPON  PARTITIONS.  177 

to  omit  the  upright  filling-in  pieces,  and  in  their  stead  to 
place  a  few  horizontal  pieces,  as  in  Fig.  58,  in  order  that  upon 
these  and  the  principal  timbers  upright  battens  may  be 
nailed  at  the  proper  distances  for  lathing.  A  partition  thus 
constructed  requires  a  little  more  space  than  others  ;  but  it 
has  the  advantage  of  insuring  greater  stability  to  the  plas- 
tering, and  also  of  preventing  to  a  good  degree  the  conver- 
sation of  one  room  from  being  overheard  in  the  adjoining 
one.  Ordinary  partitions  are  constructed  with  3x4,  3x5, 
or  4x6  inch  joists,  for  the  principal  pieces,  and  with  2x4, 
2x5,  or  2x6  filling-in  studs,  well  strutted  at  intervals  of 
about  5  feet.  When  a  partition  is  required  to  support,  in 
addition  to  its  own  weight,  that  of  a  floor  or  some  other 
burden  resting  upon  it,  the  dimensions  of  the  timbers  should 
be  ascertained,  by  applying  the  principles  which  regulate 
the  laws  of  pressure  and  those  of  the  resistance  of  timber, 
as  explained  in  the  first  part  of  this  section,  and  in  Arts.  196 
to  199  for  framed  girders.  The  following  data  may  assist  in 
calculating  the  amount  of  pressure  upon  partitions : 

White-pine  timber  weighs  from  22  to  32  pounds  per  cubic 
loot,  varying  in  accordance  with  the  amount  of  seasoning  it 
has  had.  Assuming  it  to  weigh  30  pounds,  the  weight  of 
the  beams  and  floor-plank  in  every  superficial  foot  of  the 
flooring  will  be — 

6  pounds  when  the  beams  are  3  x    8  inches,  and  placed  20  inches  from  centres 
7\      "  "  "  3  x  10          "  "          18 

9  "  "  3  x  12          "  "          16 

II  "  «  «  3XI2  "  12 

I3  "  "  "  4X12  "  "  12 

I3          «  «  «  4x14  "  14 

In  addition  to  the  beams  and  plank,  there  is  generally  the 
plastering,  of  the  ceiling  of  the  apartments  beneath,  and  some- 
times the  deafening.  Plastering  may  be  assumed  to  weigh  9 
pounds  per  superficial  foot,  and  deafening  1 1  pounds. 

Hemlock  weighs  about  the  same  as  white  pine.     A  parti- 
tion of  3x4  joists  of  hemlock,  set  12  inches  from  centres, 
therefore,  will  weigh  about  2j  pounds  per  foot  superficial 
and  when  plastered  on  both  sides,  2o£  pounds. 


CONSTRUCTION. 


ROOFS. 

202. — Roof*. — In  ancient  Norman  and  Gothic  buildings, 
the  walls  and  buttresses  were  erected  so  massive  and  firm 
that  it  was  customary  to  construct  their  roofs  without  a  tie- 
beam,  the  walls  being  abundantly  capable  of  resisting  the 
lateral  pressure  exerted  by  the  rafters.  But  in  modern 
buildings,  usually  the  walls  are  so  slightly  built  as  to  be  in- 
capable of  resisting  much  if  any  oblique  pressure  ;  hence 
the  necessity  of  care  in  constructing  the  roof  so  as  to  avoid 
oblique  and  lateral  strains.  The  roof  so  constructed,  instead 
of  tending  to  separate  the  walls,  will  bind  and  steady  them. 


FIG.  59. 


FIG.  60. 


FIG.  61. 


FIG.  62. 


FIG.  63. 


FIG.  64, 


FIG.  65. 


FIG.  66. 


FIG.  67. 


203. — Comparison  of  Roof-Tru§ses. — Designs  for  roof- 
trusses,  illustrating  various  principles  of  roof  construction, 
are  herewith  presented. 

The  designs  at  Figs.  59  to  63  are  distinguished  from  those 
at  Figs.  64  to  67  by  having  a  horizontal  tie-beam.  In  the 
latter  group,  and  in  all  designs  similarly  destitute  of  the 
horizontal  tie  at  the  foot  of  the  rafters,-  the  strains  are  much 
greater  than  in  those  having  the  tie,  unless  the  truss  be  pro- 


VARIOUS   FORMS   OF   ROOF-TRUSSES.  179 

tected  by  exterior  resistance,  such  as  may  be  afforded  by 
competent  buttresses. 

To  the  uninitiated  it  may  appear  preferable,  in  Fig.  64, 
to  extend  the  inclined  ties  to  the  rafters,  as  shown  by  the 
dotted  lines.  But  this  would  not  be  beneficial ;  on  the  con- 
trary, it  would  be  injurious.  The  point  of  the  rafter  where 
the  tie  would  be  attached  is  near  the  middle  of  its  length, 
and  consequently  is  a  point  the  least  capable  of  resisting 
transverse  strains.  The  weight  of  the  roofing  itself  tends  to 
bend  the  rafter ;  and  the  inclined  tie,  were  it  attached  to  the 
rafter,  would,  by  its  tension,  have  a  tendency  to  increase  this 
bending.  As  a  necessary  consequence,  the  feet  of  the  rafters 
would  separate,  and  the  ridge  descend. 

In  Fig.  65  the  inclined  ties  are  extended  to  the  rafters ; 
but  here  the  horizontal  strut  or  straining  beam,  located  at 
the  points  of  contact  between  the  ties  and  rafters,  counteracts 
the  bending  tendency  of  the  rafters  and  renders  these  points 
stable.  In  this  design,  therefore,  and  only  in  such  designs,  it 
is  permissible  to  extend  the  ties  through  to  the  rafters. 
Even  here  it  is  not  advisable  to  do  so,  because  of  the  in- 
creased strain  produced.  (See  Figs.  77  and  79.)  The  design 
in  Fig.  64,  66,  or  67  is  to  be  preferred  to  that  in  Fig.  65. 

204.—  Force  Diagram  :  L,oacl  upon  Each  Support.— By  a 

comparison  of  the  force  diagrams  hereinafter  given,  of  each 
of  the  foregoing  designs,  we*  may  see  that  the  strains  in  the 
trusses  without  horizontal  tie-beams  at  the  feet  of  the  rafters 
are  greatly  in  excess  of  those  having  the  tie.  In  constructing 
these  diagrams,  the  first  step  is  to  ascertain  the  reaction  of, 
or  load  carried  by,  each  of  the  supports  at  the  ends  of  the 
truss.  In  symmetrically  loaded  trusses,  the  weight  upon 
each  support  is  always  just  one  half  of  the  whole  load. 

205.— Force  Diagram  for  Trus§  in  Fig.  59. — To  obtain 
the  force  diagram  appropriate  to  the  design  in  Fig.  59,  first 
letter  the  figure  as  directed  in  Art.  195,  and  as  in  Fig.  68. 
Then  draw  a  vertical  line,  EF  (Fig.  69),  equal  to  the  weight  W 
at  the  apex  of  roof ;  or  (which  is  the  same  thing  in  effect) 
equal  to  the  sum  of  the  two  loads  of  the  roof,  one  extending 
on  each  side  of  W  half-way  to  the  foot  of  the  rafter.  Di- 


1 8o 


CONSTRUCTION. 


vide  E  F  into  two  equal  parts  at  G.  Make  G  C  and  G  D  eacTi 
equal  to  one  half  of  the  weight  N.  Now,  since  £  G  is  equal 
to  one  half  of  the  upper  load,  and  G  D  to  one  half  of  the  low- 
er load,  therefore  their  sum,  E  G  +  G  D  —  ED,  is  equal  to 
one  half  of  the  total  load,  or  to  the  reaction  of  each  support, 
E  or  F.  From  D  draw  DA  parallel  with  DA  of  Fig.  68,  and 
from  E  draw  EA  parallel  with  EA  of  Fig.  68.  The  three 
lines  of  the  triangle  A  E  D  represent  the  strains,  respectively, 
in  the  three  lines  converging  at  the  point  A  D  E  of  Fig.  68. 
Draw  the  other  lines  of  the  diagram  parallel  with  the  lines  of 


c- 


FIG.  69. 

Fig.  68,  and  as  directed  in  Arts.  195  and  197.  The  various 
lines  of  Fig.  69  will  represent  the  forces  in  the  corresponding 
lines  of  Fig.  68  ;  bearing  in  mind  (Art.  195.)  that  while  a  line 
in  the  force  diagram  is  designated  in  the  usual  manner  by  the 
letters  at  the  two  ends  of  it,  a  line  of  the  frame  diagram  is 
designated  by  the  two  letters  between  which  it  passes.  Thus, 
the  horizontal  lines  A  D,  the  vertical  lines  A  B,  and  the  in- 
clined lines  A  E  have  these  letters  at  their  ends  in  Fig.  69, 
while  they  pass  between  these  letters  in  Fig.  68. 

206.— Force  Diagram  for  Truss  in  Fig.  60. — For  this  truss 
we  have,  in  Fig.  70,  a  like  design,  repeated  and  lettered  as 
required.  We  here  have  one  load  on  the  tie-beam,  and  three 
loads  above  the  truss:  one  on  each  rafter  and  one  at  the 
ridge.  In  the  force  diagram,  Fig.  71,  make  G  H,  H  J,  and 
J  K,  by  any  convenient  scale,  equal  respectively  to  the 
weights  GH,  HJ,  and  J  K  oi  Fig.  70.  Divide  G  K  into  two 
equal  parts  at  L.  Make  LE  and  ZFeach  equal  to  one  half 
the  weight  E  F  (Fig.  70).  Then  G  Fis  equal  to  one  half  the 


FORCE   DIAGRAMS   OF  TRUSSES.  jgl 

total  load,  or  to  the  load  upon  the  support  G  (Art.  205). 
Complete  the  diagram  by  drawing  its  several  lines  parallel 
with  the  lines  of  Fig.  70,  as  indicated  by  the  letters  (see  Art. 
205),  commencing  with  G  F,  the  load  on  the  support  G  (Fig. 
70).  Draw  from  F  and  G  the  two  lines  FA  and  GA  paral- 
lel with  these  lines  in  Fig.  70.  Their  point  of  intersection 
defines  the  point  A.  From  this  the  several  points  B,  C,  and 
D  are  developed,  and  the  figure  completed.  Then  the  lines 
m  Fig.  71  will  represent  the  forces  in  the  corresponding  lines 
of  Fig.  70,  as  indicated  by  the  lettering.  (See  Art.  195.) 


K     A 


FIG.  70. 


FIG.  71. 


207. — Force  Diagram  for  Tru§§  in  Fig  61. — For  this  truss 
we  have,  in  Fig.  72,  a  similar  design,  properly  prepared  by 
weights  and  lettering ;  and  in  Fig.  73  the  force  diagram  ap- 
propriate to  it. 

In  the  construction  of  this  diagram,  proceed  as  directed 
in  the  previous  example,  by  first  constructing  N S,  the  ver- 
tical line  of  weights  ;  in  which  line  NOyOP,PQ,QR,  and  R  S 
are  made  respectively  equal  to  the  several  weights  above 
the  truss  in  Fig.  72.  Then  divide  NS  into  two  equal  parts  at  T. 
Make  7^  and  TL  each  equal  to  the  half  of  the  weight  K L. 
Make  J  K  and  L  M  equal  to  the  weights  J  K  and  L  M  of  Fig. 
72.  Now,  since  M " N  is  equal  to  one  half  of  the  weights  above 
the  truss  plus  one  half  of  the  weights  below  the  truss,  or  half  of 
the  whole  weight,  it  is  therefore  the  weight  upon  the  support 
N  (Fig.  72),  and  represents  the  reaction  of  that  support.  A 
horizontal  line  drawn  from  M  will  meet  the  inclined  line 
drawn  from  N,  parallel  with  the  rafter  A  N  (Fig.  72),  in  the 


1-82 


CONSTRUCTION. 


point^,  and  the  three  sides  of  the  triangle  A  MN,  Fig.  73,  will 
give  the  strains  in  the  three  corresponding  lines  meeting  at 
the  point  A  MN,  Fig.  72.  The  sides  of  the  triangle  HJ  5,  Fig. 


FIG.  72. 

73,  give  likewise  the  strains  in  the  three  corresponding  lines 
meeting  at  the  point  H  J  5,  Fig.  72.  Continuing  the  con- 
struction, draw  all  the  other  lines  of  the  force  diagram  parallel 


FIG.  73. 


with  the  corresponding  lines  of  Fig.  72,  and  as  directed  in 
Art.  195.  The  completed  diagram  will  measure  the  strains 
in  all  the  lines  of  Fig.  72. 


FORCE   DIAGRAMS   CONTINUED.  183 

208,— Force  Diagram  for  Truss  in  Fig.  63.— The  roof 
truss  indicated  at  Fig.  63  is  repeated  in  Fig.  74,  with  the  ad 


FIG.  74. 


FIG.  75- 


dition  of  the  lettering  required  for  the  construction  of 
force  diagram,  Fig.  75. 


1 84  CONSTRUCTION.       . 

In  this  case  there  are  seven  weights,  or  loads,  above  the 
truss,  and  three  below.  Divide  the  vertical  line  O  V  at  W 
into  two  equal  parts,  and  place  the  lower  loads  in  two  equal 
parts  on  each  side  of  W.  Owing  to  the  middle  one  of  these 
loads  not  being  on  the  tie-beam  with  the  other  two,  but  on 
the  upper  tie-beam,  the  line  G  H,  its  representative  in  the 
force  diagram,  has  to  be  removed  to  the  vertical  BJ,  and 
the  letter  M  is  duplicated.  The  line  NO  equals  half  the 
whole  weight  of  the  truss,  or  3^  of  the  upper  loads,  plus  one 
of  the  lower  loads,  plus  half  of  the  load  at  the  upper  tie- 
beam.  It  is,  therefore,  the  true  reaction  of  the  support  NO, 
and  A  N  is  the  horizontal  strain  in  the  beam  there.  It  will 
be  observed  also  that  while  H ' M  and  G  M  (Fig.  75),  which 
are  equal  lines,  show  the  strain  in  the  lower  tie-beam  at  the 
middle  of  the  truss,  the  lines  C  H  and  FG,  also  equal 'but 
considerably  shorter  lines,  show  the  strains  in  the  upper 
tie-beam.  Ordinarily,  in  a  truss  of  this  design,  the  strain  in 
the  upper  beam  would  be  equal  to  that  in  the  lower  one, 
which  becomes  true  when  the  rafters  and  braces  above  the 
upper  beam  are  omitted.  In  the  present  case,  the  thrusts  of 
the  upper  rafters  produce  tension  in  the  upper  beam  equal 
to  C  M  or  F M  of  Fig.  75,  and  thus,  by  counteracting  the 
compression  in  the  beam,  reduce  it  to  C  H  or  FG  of  the 
force  diagram,  as  shown. 


209. — Force  Diagram  for  Truss  in  Fig.  64. — The  force 
diagram  for  the  roof-truss  at  Fig.  64  is  given  in  Fig".  77, 
while  Fig.  78  is  the  truss  reproduced,  with  the  lettering 
requisite  for  the  construction  of  Fig.  77. 

The  vertical  E  F  (Fig.  77)  represents  the  load  at  the 
ridge.  Divide  this  equally  at  W,  and  place  half  the  lower 
weight  each  side  of  W,  so  that  CD  equals  the  lower 
weight.  Then  ED  is  equal  to  half  the  whole  load,  and 
equal  to  the  reaction  of  the  support  E  (Fig.  76).  The  lines 
in  the  triangle  A  D  E  give  the  strains  in  the  corresponding 
lines  converging  at  the  point  A  D  E  of  Fig.  76.  The  other 
lines,  according  to  the  lettering,  give  the  strains  in  the  cor- 
responding lines  of  the  truss.  (See  Art.  195.) 


FORCE  DIAGRAMS  CONTINUED. 


i85 


210. — Force  Diagram  for  Tru§s  in  Fig.  65. — This  truss  is 
reproduced  in  Fig.  78,  with  the  letters  proper  for  use  in  the 
force  diagram,  Fig.  79. 

< 


FIG.  79. 


Here  the  vertical   G  K,  containing  the  three  upper  loads 
GH,  HJ,  and  J K,  is  divided  equally  at   W,  and  the  lower 


1 86  CONSTRUCTION. 

load  E  F  is  placed  half  on  each  side  of  W,  and  extends  from  E 
to  F.  Then  FG  represents  one  half  of  the  whole  load  of  the 
truss,  and  therefore  the  reaction  of  the  support  G  (Fig.  78). 
Drawing  the  several  lines  of  Fig.  79  parallel  with  the  corre- 
sponding lines  of  Fig.  78,  the  force  diagram  is  complete,  and 
the  strains  in  the  several  lines  of  78  are  measured  by  the  cor- 
responding lines  of  79.  (See  Art.  195.) 

A  comparison  of  the  force  diagram  of  the  truss  in  Fig.  76 
with  that  of  the  truss  in  Fig.  78  shows  much  greater  strains 
in  the  latter,  and  we  thus  see  that  Fig.  76  or  64  is  the  more 
economical  form. 


FIG.  80. 


211. — Force  Diagram  for  Truss  in  Fig.  66. — This  truss  is 
reproduced  and  prepared  by  proper  lettering  in  Fig.  80,  and 
its  force  diagram  is  given  in  Fig.  81. 

Here  the  vertical  J M  contains  the  three  upper  loads 
JK,  KL,  and  LM.  Divide  7 J/ into  two  equal  parts  at 
G,  and  make  FG  and  G  H  respectively  equal  to  the  two 
loads  FG  and  GH  of  Fig.Zo.  Then  HJ  represents  one 
half  of  the  whole  weight  of  the  truss,  and  therefore  the  reac- 
tion of  the  support  J.  From  H  and  J  draw  lines  par- 
allel with  A  H  and  A  J  of  Fig.  80,  and  the  sides  of  the  tri- 
angle A  H  J  will  give  the  strains  in  the  three  lines  concen- 
trating in  the  point  A  H  J  (Fig.  80).  The  other  lines  of  Fig, 


EFFECT   OF  ELEVATING  THE   TIE-BEAM.  187 

8 1  are  all  drawn  parallel  with  their  corresponding  lines  in 
Fig.  80,  as  indicated  by  the  lettering.     (See  Art.  195.) 


FIG.  81. 


212 Roof-Truss:  Effect  of  Elevating  tbe  Tie-Beam.— 

From  Arts.  670,  671,  Transverse  Strains,  it  appears  that  the 


FIG.  83 


effect  of  substituting  inclined  ties  for  the  horizontal  tie  at 
feet  of  rafters  is— 


1  88  CONSTRUCTION. 


in  which  P  represents  half  the  weight  of  the  whole  truss  and 
the  load  upon  it  ;  a  +  &  =  height  of  the  truss  at  middle  above 
a  horizontal  line  drawn  at  the  feet  of  the  rafters  ;  a  equals 
the  height  from  this  line  to  the  point  where  the  two  inclined 
ties  meet  ;  b,  the  height  thence  to  the  top  of  the  truss  ;  and 
F,  the  additional  vertical  strain  at  the  middle  of  the  truss 
due  to  elevating  the  tie  from  a  horizontal  line. 

Examples  are  given  to  show  that  when  the  elevation  of 
the  tie  equals  i  of  the  whole  height,  the  vertical  strain  there- 
by induced  is  equal  to  a  weight  which  equals  \  of  half  the 
whole  load  ;  and  that  when  the  elevation  equals  half  the 
whole  height,  the  vertical  strain  is  equal  to  half  the  whole 
load.  This  is  the  strain  in  the  vertical  rod  at  middle.  The 
strains  in  the  rafters  and  inclined  ties  are  proportionately 
increased. 

213.  —  Planning  a  Roof.  —  In  designing  a  roof  for  a  build- 
ing, the  first  point  requiring  attention  is  the  location  of  the 
trusses.  These  should  be  so  placed  as  to  secure  solid  bear- 
ings upon  the  walls  ;  care  being  taken  not  to  place  either  of 
the  trusses  over  an  opening,  such  as  those  for  windows  or 
doors,  in  the  wall  below.  Ordinarily,  trusses  are  placed  so 
as  to  be  centrally  over  the  piers  between  the  windows  ;  the 
number  of  windows  consequently  ruling  in  determining  the 
number  of  trusses  and  their  distances  from  centres.  This 
distance  should  be  from  ten  to  twenty  feet  ;  fifteen  feet  apart 
being  a  suitable  medium  distance.  The  farther  apart  the 
trusses  are  placed,  the  more  they  will  have  to  carry  ;  not 
only  in  having  a  larger  surface  to  support,  but  also  in  that 
the  roof-timbers  will  be  heavier  ;  for  the  size  and  weight,  of 
the  roof-beams  will  increase  with  the  span  over  which  they 
have  to  reach. 

In  the  roof-covering  itself,  the  roof-planking  may  be  laid 
upon  jack-rafters,  carried  by  purlins  supported  by  the 
trusses  ;  or  upon  roof-beams  laid  directly  upon  the  back  of 
the  principal  rafters  in  the  trusses.  In  either  case,  proper 


LOAD   UPON   ROOFS.  189 

struts  should  be  provided,  and  set  at  proper  intervals  to  re- 
sist the  bending  of  the  rafter.  Jn  case  purlins  are  used,  one  of 
these  struts  should  be  placed  at  the  location  of  each  purlin. 

The  number  of  these  points  of  support  rules  largely  in 
determining  the  design  for  the  truss,  thus : 

For  a  short  span,  where  the  rafter  will  not  require  sup- 
port at  an  intermediate  point,  Fig.  59  or  64  will  be  proper. 

For  a  span  in  which  the  rafter  requires  supporting  at  one 
intermediate  point,  take  Fig.  60,  65,  or  66. 

For  a  span  with  two  intermediate  points  of  support  for 
the  rafter,  take  Fig.  61  or  67. 

For  a  span  with  three  intermediate  points,  take  Fig.  63. 

Generally,  it  is  found  convenient  to  locate  these  points  of 
support  at  nine  to  twelve  feet  apart.  They  should  be  suffi- 
ciently close  to  make  it  certain  that  the  rafter  will  not  be  sub- 
ject to  the  possibility  of  bending. 

214.— Load  upon  Roof-Tru§§.— In  constructing  the  force 
diagram  for  any  truss,  it  is  requisite  to  determine  the  points 
of  the  truss  which  are  to  serve  as  points  of  support  (see 
Figs.  70,  72,  etc.),  and  to  ascertain  the  amount  of  strain,  or 
loading,  which  will  occur  at  every  such  point. 

The  points  of  support  along  the  rafters  will  be  required 
to  sustain  the  roofing  timbers,  the  planking,  the  slating,  the 
snow,  and  the  force  of  the  wind.  The  points  along  the  tie- 
beam  will  have  to  sustain  the  weight  of  the  ceiling  and  the 
flooring  of  a  loft  within  the  roof,  if  there  be  one,  together 
with  the  loading  upon  this  floor.  The  weight  of  the  truss 
itself  must  be  added  to  the  weight  of  roof  and  ceiling. 

215,-lLoacl  on  Roof  per  Superficial  Foot — In  any  im- 
portant work,  each  of  the  items  in  Art.  214  should  be  care- 
fully estimated,  in  making  up  the  load  to  be  carried.  For 
ordinary  roofs,  the  weights  may  be  taken  per  foot  superficial, 
as  follows : 

Slate,  about     7-0    pounds. 

Roof-plank, 

Roof-beams  or  jack-rafters,  2  -  3 

In  all,  I2         pounds. 


CONSTRUCTION. 

This  is  for  the  superficial  foot  of  the  inclined  roof.  For  the 
foot  horizontal,  the  augmentation  of  load  due  to  the  angle  of 
the  roof  will  be  in  proportion  to  its  steepness.  In  ordinary 
cases,  the  twelve  pounds  of  the  inclined  surface  will  not  be 
far  from  fifteen  pounds  upon  the  horizontal  foot. 
For  the  roof-load  we  may  take  as  follows  : 

Roofing,  about  15  pounds. 

Roof-truss,           "  5 

Snow,                   "  20          " 

Wind,                    "  10          «. 

Total  on  roof,         50    pounds 

per  square  foot  horizontal. 

This  estimate  is  for  a  roof  of  moderate  inclination,  say 
one  in  which  the  height  does  not  exceed  J  of  the  span. 
Upon  a  steeper  roof  the  snow  would  not  gather  so  heavily, 
but  the  wind,  on  the  contrary,  would  exert  a  greater  force. 
Again,  the  wind  acting  on  one  side  of  a  roof  may  drift  the 
snow  from  that  side,  and  perhaps  add  it  to  that  already 
lodged  upon  the  opposite  side.  These  two,  the  wind  and 
the  snow,  are  compensating  forces.  The  action  of  the  snow 
is  vertical :  that  of  the  wind  is  horizontal,  or  nearly  so.  The 
power  of  the  wind  in  this  latitude  is  not  more  than  thirty 
pounds  upon  a  superficial  foot  of  a  vertical  surface  ;  except, 
perhaps,  on  elevated  places,  as  mountain-tops  for  example, 
where  it  should  be  taken  as  high  as  fifty  pounds  per  foot  of 
vertical  surface. 

216. — Load  upon  Tic-Beam. — The  load  upon  the  tie- 
beam  must  of  course  be  estimated  according  to  the  require- 
ments of  each  case.  If  the  timber  is  to  be  exposed  to  view, 
the  load  to  be  carried  will  be  that  only  of  the  tie-beam  and 
the  timber  struts  resting  upon  it.  If  there  is  to  be  a  ceiling 
attached  to  the  tie-beam,  the  weight  to  be  added  will  be  in 
accordance  with  the  material  composing  the  ceiling.  If  of 
wood,  it  need  not  weigh  more  than  two  or  three  pounds  per 
foot.  If  of  lath  and  plaster,  it  will  weigh  about  nine  pounds  ; 
and  if  of  iron,  from  ten  to  fifteen  pounds,  according  to  the 


WEIGHT   UPON    ROOFS,    IN   DETAIL.  191 

thickness  of  the  metal.  Again,  if  there  is  to  be  a  loft  in  the 
roof,  the  requisite  flooring  may  be  taken  at  five  pounds,  and 
the  load  upon  the  floor  at  from  twenty-five  to  seventy 
pounds,  according  to  the  purpose  for  which  it  is  to  be  used. 

217 — Roof- Weights  in  B»eta51.— The  load  to  be  sustained 
by  a  roof-truss  has  been  referred  to  in  the  previous  three 
articles  in  general  terms.  It  will  now  be  treated  more  in 
detail.  But  first  a  few  words  regarding  fehe  slope  of  the 
roof.  In  a  severe  climate,  roofs  ought  to  be  constructed 
steeper  than  in  a  milder  one,  in  order  that  snow  may  have  a 
tendency  to  slide  off  before  it  becomes  of  sufficient  weight 
to  endanger  the  safety  of  the  roof.  In  selecting  the  material 
with  which -the  roof  is  to  be  covered,  regard  should  be  had 
to  the  requirements  of  the  inclination :  slate  and  shingles 
cannot  be  used  safely  on  roofs  of  small  rise.  The  smallest 
inclination  of  the  various  kinds  of  covering  is  here  given, 
together  with  the  weight  per  superficial  foot  of  each. 


MATERIAL. 


Least  Inclination. 


Weight  upon  a 
square  foot. 


Tin  ;  

F 

se  i  inch  to 

a  f  c 

or 

1  to  ii  1 

3S. 

CopDcr 

i     " 

i    to  i^ 

v^uppci  

Lead  ....             .                       . 

2  inches 

4    to  7 

< 

Zinc  

3      " 

i£  to  2 

M 

Short  pine  shingles  

5      " 

l£  tO  2 

w 

Long  cypress  shingles     .         .        

'  6      " 

2    to  3 

« 

Slate  

6      " 

5    to  9 

< 

The  weight  of  the  covering  as  here  estimated  includes 
the  weight  of  whatever  is  used  to  fix  it  in  place,  such  as 
nails,  etc.  The  weight  of  that  which  the  covering  is  laid 
upon,  such  as  plank,  boards,  or  lath,  is  not  included.  The 
weight  of  plank  is  about  3  pounds  per  foot  superficial ;  of 
boards,  2  pounds  ;  and  lath,  about  half  a  pound. 

Generally,  for  a  slate  roof,  the  weight  of  the  covering,  in- 
cluding plank  and  jack-rafters,  amounts  to  about  12  pounds, 
as  stated  in  Art.  215  ;  but  in  every  case,  the  weight  of  each 
article  of  the  covering  should  be  estimated,  and  the  full  load 
ascertained  by  summing  up  these  weights. 


1 92  CONSTRUCTION. 

218. — Load  per  Foot  Horizontal. — The  weight  ot  the 
covering  as  referred  to  in  the  last  article  is  the  weight  per 
foot  on  the  inclined  surface  ;  but  it  is  desirable  to  know  how 
much  per  foot,  measured  horizontally,  this  is  equal  to.  The 
horizontal  measure  of  one  foot  of  the  inclined  surface  is 
equal  to  the  cosine  of  the  angle  of  inclination.  Then,  to  ob- 
tain the  inclined  measure  corresponding  to  one  foot  horizon- 
tal, we  have — 


cos.  :  I  :  :  p  :  C  = 

cos. 

where  /  represents  the  pressure  on  a  foot  of  the  inclined 
surface,  and  C  the  weight  of  so  much  of  the  inclined  cover- 
ing as  corresponds  to  one  foot  horizontal.  The  cosine  of  an 
angle  is  equal  to  the  base  of  the  right-angled  triangle  divided 
by  the  hypothenuse  (see  Trigonometrical  Terms,  Art.  474), 
which  in  this  case  is  half  the  span  divided  by  the  length  of 

the  rafter,  or — -. ,  where  s  is  the  span,  and  /  the  length  of  the 
rafter.  Hence,  the  load  per  foot  horizontal  equals — 

p         p  _2  Ip 

^c^sT^T1    ~T~  (92.) 

2/ 

or,  twice  the  pressure  per  foot  of  inclined  surface  multiplied 
by  the  length  of  the  rafter  and  divided  by  the  span,  both  in 
feet,  will  give  the  weight  per  foot  measured  horizontally. 

219. — WeigBit  of  Tru§§. — The  weight  of  the  framed  truss 
will  be  in  proportion  to  the  load  and  to  the  span.  This,  for 
the  weight  upon  a  foot  horizontal,  will  about  equal— 

T  —  0-077  Cs\ 

which  equals  the  weight  in  pounds  per  foot  horizontal  to  be 
allowed  for  a  wooden  truss  with  iron  suspension-rods  and  a 
horizontal  tie-beam,  near  enough  for  the  requirements  of  our 
present  purpose  ;  where  s-  equals  the  length  or  span  -of  the 


EFFECT  OF   WIND  ON   ROOFS.  193 

truss,  and  C  the  weight  per  foot  horizontal  of  the  roof  cover- 
ing, as  in  equation  (92.).  Substituting  for  Cits  value,  as  in 
(92.),  we  have — 

T=  0-0077^; 
or — 

T  =  0-0154  lp\  (93.) 

which  equals  the  weight  in  pounds  per  foot  horizontal  to 
be  allowed  for  the  truss. 

220. — Weight  of  Snow  on  Rooffc. — The  weight  of  snow 
will  be  in  proportion  to  the  depth  it  acquires,  which  will  be 
in  proportion  to  the  rigor  of  the  climate  of  the  place  where 
the  building  is  to  be  erected.  Upon  roofs  of  ordinary  incli- 
nation, snow,  if  deposited  in  the  absence,  of  wind,  will  not 
slide  off ;  at  least  until  after  it  has  acquired  some  depth,  and 
then  the  tendency  to  slide  will  be  in  proportion  to  the  angle 
of  inclination.  The  weight  of  snow  may  be  taken,  therefore, 
at  its  weight  per  cubic  foot  (8  pounds)  multiplied  by  the 
depth  it  is  usual  for  it  to  acquire.  This,  in  the  latitude  of 
New  York,  may  be  taken  at  about  2-J-  feet.  Its.  weight 
would,  therefore,  be  20  pounds  per  foot  superficial,  meas- 
ured horizontally. 

221. — Effect  of  Wind  on  Roof*. — The  direction  of  wind 
is  horizontal,  or  nearly  so,  when  unobstructed.  Precipitous 
mountains  or  tall  buildings  deflect  the  wind  considerably 
from  its  usual  horizontal  direction.  Its  power  usually  does 
not  exceed  30  pounds  per  superficial  foot  except  on  ele- 
vated places,  where  it  sometimes  reaches  50  pounds  or  more. 
This  is  the  pressure  upon  a  vertical  surface  ;  roofs,  however, 
generally  present  to  the  wind  an  inclined  surface.  The  ef- 
fect of  a  horizontal  force  on  an  inclined  surface  is  in  pro- 
portion to  the  sine  of  the  angle  of  inclination  ;  the  direction 
of  this  effect  being  at  right  angles  to  the  inclined  surface. 
The  force  thus  acting  may  be  resolved  into  forces  acting  in 
two  directions— the  one  horizontal,  the  other  vertical ;  the 
former  tending,  in  the  case  of  a  roof,  to  thrust  aside  the  walls 


194 


CONSTRUCTION. 


on  which  the  roof  rests,  and  the  latter  acting  directly  on  the 
materials  of  which  the  roof  is  constructed — this  latter  force 
being  in  proportion  to  the  sine  of  the  angle  of  inclination 
multiplied  by  the  cosine.  This  will  be  made  clear  by  the 

following  explanation.  Re- 
ferring to  Fig.  83,  let  D  KE 
be  the  angle  of  inclination 
of  the  roof,  D  E  being  equal 
to  one  foot.  Bisect  DK  at 
A  ;  draw  A  L  parallel  with 
FIG.  83.  EK\  make  A  L  equal  to  the 

horizontal  pressure  of  the  wind  upon  one  foot  superficial  of 
a  vertical  plane.  Draw  A  C  perpendicular  to  D K,  and  LF 
parallel  with  A  C  from  F  draw  FC  parallel  with  EK\  draw 
A  B  parallel  with  D  E.  The  sides  of  the  triangle  LA  F  rep- 
resent the  three  several  forces  in  equilibrium  :  LA  is  the 
force  of  the  wind  ;  L  F  is  the  pressure  upon  the  roof ;  and 
A  F  is  the  force  with  which  the  wind  moves  on  up  the  roof 
towards  D.  Now,  to  find  the  relation  of  the  force  of  the 
wind  to  the  strain  produced  by  it  in  the  direction  A  C,  we 
have — 


rad. 


rad.  :  sin.  \\FC\AC\ 
F  C  =  LA;  therefore— 
:  sin.  :  :  L  A  :  A  C  —  L  A  sin.  ; 
AC  =  Fsin.-t 


or,  the  strain  perpendicular  to  the  surface  of  the  roof  equals 
the  force  of  the  wind  multiplied  by  the  sine  of  the  angle  of 
inclination.  When  A  C  represents  this  strain,  then,  of  the 
two  forces  referred  to  above,  B  C  represents  the  horizontal 
force,  and  A  B  the  vertical  force.  To  obtain  this  last  force, 
we  have — 

rad.  :  cos.  \\AC\AB. 
Putting  for  A  C  its  value  as  above,  we  have — 

rad.  :  cos.  :  :  /^sin.  :  A  B  =  F  sin.  cos.; 
V  =  F  sin.  cos. ; 


FORCE   OF  THE   WIND.  195 

or,  the  vertical  effect  is  equal  to  the  product  of  the  force  of 
the  wind  upon  a  superficial  foot  into  the  sine  and  the  cosine 
of  the  angle  of  inclination.  This  result  is  that  which  is  due 
to  the  pressure  of  the  wind  upon  so  much  of  the  inclined 
surface  as  is  covered  by  one  square  foot  of  a  vertical  sur- 
face. The  wind,  acting  horizontally  through  one  foot  super- 
ficial of  vertical  section,  acts  on  an  area  of  inclined  surface 
equal  to  the  reciprocal  of  the  sine  of  inclination,  and  the 
horizontal  measurement  of  this  inclined  surface  is  equal  to 
the  cosine  of  the  angle  of  inclination  divided  by  the  sine. 
This  may  be  illustrated  from  Fig.  83,  thus— 

sin.  :  rad.  \\DE\DK. 

D  E  equals  I  foot ;  therefore — •  .+•. 

sin.  :  rad.  :  :  I  :  D  K  =  — !—  ; 

sin. 

or,  the  surface  acted  upon  by  one  square  foot  of  sectional 
area  equals  the  reciprocal  of  the  sine  of  the  angle  of  incli- 
nation. Again,  the  horizontal  measure  of  this  inclined  sur- 
face may  be  obtained  thus— 

cos. 

sin. :  cos.  :  :  D  E  :  K  E  =  — —  ; 

sin. 

or,  KE,  the  horizontal  measurement,  equals  the  cosine  of  the 
angle  of  inclination  divided  by  the  sine. 

In   tile   figure,  make   K  G  equal   to  one  foot ;    then  we 
have — 

K  E  :  KG  :  :   V  '.    W\ 

in  which  V,  as  above,  represents  the  vertical  pressure  due  to 
the  wind  acting  upon  the  surface  KD,  and  W  the  vertical 
pressure  due  to  the  wind  acting  upon  the  surface  KH,  or 
so  much  as  covers  KG,  one  foot  horizontal. 

Now  we  have,  as  above,  K  E  equal  to  — -— ,  K.  G  =  i,  and 


196  CONSTRUCTION. 

V  —  F  sin.  cos.     Substituting  these  values,  we  have,  instead 
of  the  above  proportion- 

cos.  _   . 

—  —  :  i  :  :  F  sin.  cos.  :  W\ 

sin. 

from  which  — 

'f±?  (94.) 


sn. 

or,  the  vertical  effect  of  the  wind  upon  so  much  of  the  roof 
as  covers  each  square  foot  horizontal,  is  equal  to  the  pro- 
duct of  the  force  of  the  wind  per  square  foot  into  the  square 
of  the  sine  of  the  angle  of  inclination. 

Example.  —  When  the  force  of  the  wind  upon  a  square 
foot  of  vertical  surface  is  30  pounds,  what  will  be  the  verti- 
cal effect  per  square  foot  horizontal  upon  a  roof  the  inclina- 
tion of  which  is  26°  33',  or  6  inches  to  the  foot? 

Here  we  have  F  =  30,  and  the  sine  of  26°  33'  is  0-44698  ; 
therefore— 

W—  30  x  0-44698  2  =  5-9937- 
This  is  conveniently  solved  by  logarithms  ;  thus— 

log.  30  =  1^-4771213 
0-44698  =  9-6502868 
0-44698  =  9^-6502868 

5-9937     =  0-7776949 

or,  the  vertical  effect  is  (5  -9937,  or)  6  pounds. 

The  form  of  equation  (94.)  may  be  changed  ;  for,  in  a  right- 
angled  triangle,  the  sine  of  the  angle  at  the  base  is  equal  to 
the  perpendicular  divided  by  the  hypothenuse  ;  which,  in 
the  case  of  a  roof,  is  the  height  divided  by  the  length  of  the 
rafter;  or  — 

height        h 
Sme  =  TaTteF  =  2 


LOAD   UPON   ROOFS.  197 

Therefore,  equation  (94.)  may  be  changed  to  — 


(950 


or,  the  vertical  effect  upon  each  square  foot  of  a  roof  is  equal 
to  the  product  of  the  force  of  the  wind  per  foot  into  the 
square  of  the  height  of  the  roof  at  the  ridge,  divided  by  the 
square  of  the  length  of  the  rafter  (the  height  and  length  both 
in  feet.) 

Example.  —  When  the  force  of  the  wind  is  30  pounds,  the 
height  of  the  roof  10  feet,  and  the  length  of  the  rafter  22-36 
feet,  what  will  be  the  vertical  effect  of  the  wind  ?  Here  we 
have  F  ~  30,  h  =  10,  and  /  =  22-36  ;  and— 


222.  —  Total  Load  per  Foot  Horizontal.  —  The  various 
items  comprising  the  total  load  upon  a  roof  are  the  cover- 
ing, the  truss,  the  wind,  snow,  the  plastering  or  other  kind 
of  ceiling,  and  the  load  which  may  be  deposited  upon  a  floor 
formed  in  the  roof  ;  or,  the  total  load  — 

M=  C+T  +  W+S  +  P+L. 

The  value  per  foot  horizontal  for  these  has  been  found  as 
follows:  C=^;  T=  0-0154  //;  W=F^.  For  5  the 

value  must  be  taken  according  to  circumstances,  as  in  Art. 
220.  So,  also,  the  value  of  P  and  L  are  to  be  assigned  as 
required  for  each  particular  case,  as  in  Art.  216.  The  total 
load,  therefore,  with  these  substitutions,  will  be— 

M  = 


which  reduces  to — 

M  =  lp  (-  +  0-0154)  +  F^t  \-S  +  P+L;     (96.) 

*  S  /  l> 


198  CONSTRUCTION. 

in  which  /  is  the  length  of  the  rafter ;  /  is  the  weight  of  the 
covering  per  foot  superficial,  including  the  roof  boards  or 
slats,  the  jack-rafters,  etc. ;  s  is  the  span  of  the  roof ;  ~h  is  the 
vertical  height  above  a  horizontal  line  passing  through  the 
feet  of  the  rafters ;  F  is  the  force  of  the  wind  per  square  foot 
against  a  vertical  surface  ;  5  is  the  weight  of  snow  per 
square  foot  horizontal ;  P  is  the  weight  per  superficial  foot 
of  the  ceiling  at  the  tie-beam  ;  and  L,  the  load  per  superficial 
foot  in  the  roof,  including  weight  of  flooring  and  floor- 
timbers.  The  dimensions,  s,  /,  and  /i,  are  each  in  feet ;  the 
weight  of  /,  F,  5,  P,  and  L  are  each  in  pounds.  The  value 
of/  is  for  a  square  foot  of  the  inclined  surface. 

223. — Strain§  in  Roof-Timbers  Computed.— The  graphic 
method  of  obtaining  the  strains,  as  shown  in  Arts.  205  to  211, 
is,  for  its  conciseness  and  simplicity,  to  be  preferred  to  any 
other  method ;  yet,  on  some  accounts,  the  method  of  obtain- 
ing the  strains  by  the  parallelogram  of  forces  and  by  arith- 
metical computations  will  be  found  useful,  and  will  now  be 
referred  to. 

By  the  parallelogram  of  forces,  the  weight  of  the  roof  is 
in  proportion  to  the  oblique  thrust  or  pressure  in  the  axis  of 
the  rafter  as  twice  the  height  of  the  roof  is  to  the  length  of 
the  rafter ;  or — 

R:  F::  2  A:  /; 
or,  transposing — 

2&:i::R:Y=j£;  (97-) 

where  F  equals  the  pressure  in  the  axis  of  the  rafter,  and  R 
the  weight  of  one  truss  and  its  load.  Again,  the  weight  of 
the  roof  is  in  proportion  to  the  horizontal  thrust  in  the  tie- 
beam  as  twice  the  height  of  the  roof  is  to  half  the  span  ;  or — 


2 


or,  transposing — 

2/i:S-::R:H=~',  (98.) 


THE   STRAINS   SHOWN  GEOMETRICALLY.  199 

where  H  equals  the  horizontal  thrust  in  the  tie-beam.  To 
obtain  R,  the  weight  of  the  roof,  multiply  M,  the  load  per 
foot,  as  in  equation  (96.),  by  s,  the  span,  and  by  c,  the  dis- 
tance from  centres  at  which  the  trusses  are  placed  ;  or 

R  =  M  c  s. 
With  this  value  of  R  substituted  for  it,  we  have — 

K=  —  — ' 

and — 

TT      M  c  s*  ,       . 

H  = T—  ;  (loo.) 

4  It 

in  which  F  equals  the  strain  in  the  axis  of  the  rafter,  and  H 
the  strain  in  the  tie-beam.  These  are  the  greatest  strains 
in  the  rafter  and  tie-beam.  At  certain  parts  of  these  pieces 
the  strains  are  less,  as  will  be  shown  in  the  next  article. 

224. — Strains  in  Roof-Timber§  Shown  Geometrically. — 

The  pressure  in  each  timber  may  be  obtained  as  shown  in 
Fig.  84,  where  A  B  represents  the  axis  of  the  tie-beam,  A  C 
the  axis  of  the  rafter,  D  E  and  F  B  the  axes  of  the  braces, 
and  DG,  FE,  and  C  B  the  axes  of  the  suspension-rods.  In 
this  design  for  a  truss,  the  distance  A  B  is  divided  into  three 
equal  parts,  and  the  rods  located  at  the  two  points  of  division, 
G  and  E.  By  this  arrangement  the  rafter  A  £7 is  supported  at 
equidistant  points,  D  and  F.  The  point  D  supports  the  rafter 
for  a  distance  extending  half-way  to  A  and  half-way  to  F,  and 
the  point  F  sustains  half-way  to  D  and  half-way  to  C.  Also, 
the  point  C  sustains  half-way  to  F,  and,  on  the  other  rafter, 
half-way  to  the  corresponding  point  to  F.  And  because  these 
points  of  support  are  located  at  equal  distances  apart,  there- 
fore the  load  on  each  is  the  same  in  amount.  On  D  G  make 
Da  equal  by  any  decimally  divided  scale  to  the  number  of 
hundreds  of  pounds  in  the  load  on  D,  and  draw  the  parallel- 
ogram abDc.  Then,  by  the  same  scale,  Db  represents 
(Art.  71)  the  pressure  in  the  axis  of  the  rafter  by  the  load  at 


2OO 


CONSTRUCTION, 


FIG.  84. 


STRAINS   IN   A.  TRUSS.  2OI 

D\  also,  DC  the  pressure  in  the  brace  D E.  Draw  cd  hori- 
zontal ;  then  D  d  is  the  vertical  pressure  exerted  by  the  brace 
D  E  at  E.  The  point  F  sustains,  besides  the  common  load 
represented  by  D  a,  also  the  vertical  pressure  exerted  by  the 
brace  D  E ;  therefore,  make  Fe  equal  to  the  sum  of  D  a  and 
Dd,  and  draw  the  parallelogram  F gef.  Then  Fg,  meas- 
ured by  the  scale,  is  the  pressure  in  the  axis  of  the  rafter 
caused  by  the  load  at  F,  and  F  f  is  the  load  in  the  axis  of  the 
brace  FB.  Draw  fh  horizontal ;  then  Fh  is  the  vertical 
pressure  exerted  by  the  brace  Ffiat  B.  The  point  C,  besides 
the  common  load  represented  by  D  a,  sustains  the  vertical 
pressure  Fh  caused  by,  the  brace  FB,  and  a  like  amount 
from  the  corresponding  brace  on  the  opposite  side.  There- 
fore, make  Cj  equal  to  the  sum  of  Da  and  twice  Fit,  and 
draw  jk  parallel  to  the  opposite  rafter.  Then  Ck  is  the 
pressure  in  the  axis  of  the  rafter  at  C.  This  is  not  the  only 
pressure  in  the  rafter,  although  it  is  the  total  pressure  at  its 
head  C.  At  the  point  F,  besides  the  pressure  C  k,  there  is 
F g.  At  the  point  D,  besides  these  two  pressures,  there  is 
the  pressure  D  b.  At  the  foot,  at  A,  there  is  still  an  addi- 
tional pressure  ;  for  while  the  point  D  sustains  the  load  half- 
way to  F  and  half-way  to  A,  the  point  A  sustains  the  load 
half-way  to  D.  This  load  is,  in  this  case,  just  half  the  load 
at  D.  Therefore  draw  A  m  vertical,  and  equal,  by  the  scale, 
to  half  of  Da.  Extend  CA  to/;  draw  ml  horizontal. 
Then  A  I  is  the  pressure  in  the  rafter  at  A  caused  by  the 
weight  of  the  roof  from  A  half-way  to  D.  Now  the  total  of 
the  pressures  in  the  rafter  is  equal  to  the  sum  of  A  1+  D  b  + 
Fg  added  to  C  k.  Therefore  make  kn  equal  to  the  sum  of 
A  l+Db  +  F g,  and  draw  no  parallel  with  the  opposite  raf- 
ter, and  nj  horizontal.  Then  Co,  measured  by  the  same 
scale,  will  be  found  equal  to  the  total  weight  of  the  roof  on 
both  sides  of  C  B.  Since  Da  represents  s,  the  portion  of  the 
weight  borne  by  the  point  D,  therefore  Co,  representing  the 
whole  weight  of  the  roof,  should  equal  six  times  Da,  as  it 
does,  because  D  supports  just  one  sixth  of  the  whole  load. 
Since  C  n  is  the  total  oblique  thrust  in  the  axis  of  the  rafter 
at  its  foot,  therefore  nj  is  the  horizontal  thrust  in  the  tie- 
beam  at  A. 


2O2  CONSTRUCTION. 

225.— Application  of  the  Geometrical  System  of  Strains.— 

The  strains  in  a  roof-truss  can  be  ascertained  geometrically, 
as  shown  in  Art.  224.  To  make  a  practical  application  of 
the  results,  in  any  particular  case,  it  is  requisite  first  to  as- 
certain the  load  at  the  head  of  each  brace,  as  represented  by 
the  line  D  a,  Fig.  84.  The  load  corresponding  to  any  part 
of  the  roof  is  equal  to  the  product  of  the  superficial  area  of 
that  particular  part  (measured  horizontally)  multiplied  by 
the  weight  per  square  foot  of  the  roof.  Or,  when  M  equals 
the  weight  per  square  foot,  c  the  distance  from  centres  at 
which  the  trusses  are  placed,  and  n  the  horizontal  distance 
between  the  heads  of  the  braces,  then  the  total  load  at  the 
head  of  a  brace  is  represented  by— 

N—Mcn.  (101.) 

The  value  of  M  is  given  in  general  terms  in  equation  (96.). 
To  show  its  actual  value,  let  it  be  required  to  find  the  weight 
per  square  foot  upon  a  root  52  feet  span  and  13  feet  high  at 
middle  ;  or  (Fig.  84),  where  A  B  equals  half  the  space,  or  26 
feet,  and  C  B  13  feet,  then  ^4  C,  the  length  of  the  rafter,  will 
be  26-069,  nearly.  And  where  the  weight  of  covering  per 
square  foot,  on  the  inclination,  is  12  pounds,  the  force  of  the 
wind  against  a  vertical  plane  is  30  pounds ;  the  weight  of 
snow  per  foot  horizontal  is  20  pounds ;  the  weight  of  the 
plastering  forming  the  ceiling  at  the  tie-beam  is  9  pounds ; 
and  the  load  in  the  roof  is  nothing  ; — with  these  quantities 
substituted,  equation  (96.)  becomes— 

M  —  2Q'o6gx  12  (--  4-  0-0154)  4-  30 — 1^ —  +  20  49  4-0  ; 
\52  /         29-069' 

M  =  (29-069  X   12  X  0-05386)  4  (30  x  0-2)  +  20  4-  9; 

M  =  18-788  4-6  +  29  =  53-788; 

or,  say,  53-8  pounds.  Then  if  <r,  the  distance  from  centres 
between  trusses,  is  10  feet,  and  ;/,  the  distance  between 
braces,  is  one  third  of  A  B,  Fig.  84,  or  2/  =  8f ,  the  total  load 
at  the  head  of  a  brace  will  be,  as  per  equation  (101.) — 

N—  53-8  x  10  x  8f  =  4663; 


TABLE   OF   STRAINS.  203 

or,  say,  4650  pounds.  Now,  by  any  decimally  divided  scale, 
make  D  a,  Fig.  84,  equal  to  46^  parts  of  the  scale ;  this  being 
the  number  of  hundreds  of  pounds  contained  in  the  weight 
at  D,  as  above.  Then,  by  the  same  scale,  the  several  lines 
in  the  figure  drawn  as  before  shown  will  be  found  to  repre- 
sent respectively  the  weights  here  set  opposite  to  them,  as 
follows  : 

D  d  —  da  —  he  —  23^,  and  represents  2325  pounds; 

"  4650 

"  5200         " 

F  e  =  D  a  +  D  d  —  69$                     "  6975 

Ff  =  65|                                               «  6575 

Cj  —  3  D  a  —  139-1-                           "  13950 

CK  =  $  D  b  =  \$6                           "  15600 

C  n  =  C  k +-  Fg+  D  b  +  A  1=  312    "  31200         " 
Cn^Ck  +  $Db=6  Db=2  Ck 

=  312                                       "  31200 

Nj  —  C  o  —  6  D  a  —  6^  4.6%  —  279  "  27900         *' 

It  should  be  observed  here  that  the  equality  of  the  lines  nj 
and  Co  is  a  coincidence  dependent  upon  the  relation  which 
in  this  particular  case  the  line  CB  happens  to  bear  to  the 
line  A  B  ;  A  B  being  equal  to  twice   C  B.     And  so  of  some 
other  lines  in  the  figure.     If  the  inclination  of  the  roof  were 
made  greater  or  less,  the  equality  of  the  lines  referred  to 
would  disappear.     It  should  also  fye  observed  that  the  strains 
above  found  are  not  quite  exact ;  they  are,  however,  correct 
to  within  a  fraction  of  a  hundred  pounds,  which  is  a  suffi- 
ciently near  approximation  for  the  purpose  intended.     From 
the  results  obtained  above,  we  ascertain  that  the  strain  in 
the  rafter,  from  F  to  C,  is  represented   by  C  K,  and  is  equal 
to  15,600  pounds ;  while  the  strain  at  the  foot  of  the  rafter, 
from  A  to  D,  is  represented  by  C  n,  and  equals  3 1,200  pounds, 
or  double  that  which  is  at  the  head  of  the  rafter.     We  ascer- 
tain, also,  that  the  maximum  strain  in  the  tie-beam,  repre- 
sented by  11  j,  is  27,900  pounds;  that  that  in  the  brace  D  E, 
represented  by   DC,  is  5200  pounds;  and  that  that  in  the 
brace  F  B,  represented  by  Ff,  is  6575   pounds.     The  strain 


204  CONSTRUCTION. 

in  the  vertical  rod  D  G  is  theoretically  nothing.  There  is, 
however,  a  small  strain  in  it,  for  it  has  to  carry  a  part  of  the 
tie-beam  and  so  much  of  the  ceiling  as  depends  for  support 
upon  that  part.  But  the  manner  of  locating  the  weights, 
adopted  in  this  article,  does  not  recognize  any  load  located 
at  the  point  G.  This  is  an  objection  to  this  system,  but  it 
is  not  material. 

For  a  recognition  of  weights  at  the  tie-beam,  see  Arts. 

205  to  211.     The  load  at  G  may  be  found  by  obtaining  the 
product  of  the  surface  carried   into   the  weight  per  foot  of 
the    ceiling;    or,  say,    10  c  n  =  10  x  lox  8f  =  867    pounds. 
The  load  to  be  carried  by  the  rod  F  E  is  shown  at  D  d=  he, 
which  above  is  found  to  be  2325  pounds.     To  this  is  to  be 
added  867  pounds  for  the  ceiling  at  E,  as  before  found  for  the 
ceiling  at  G\  or,  together,  3192  pounds.     The  central  rod 
C  B  has  to  carrv  the  two  loads  brought  to  B  by  the  two 
braces  footed  there  ;  and  also  the  weight  of  the  ceiling  sup- 
ported by  B.     The  vertical  strain  from  the  brace  F  B  is  rep- 
resented   at  Fh,  and    equals  4650   pounds  ;    therefore,  the 
total  load  on  CB  is  4650  +  4650  -f  867  =  10,167  pounds. 

226.  —  Roof-Timber*  :  the  Tie-Beam.  —  The  roof-timbers 
comprised  in  the  truss  shown  in  Fig.  84  are  the  rafters, 
tie-beam,  two  braces,  and  three  rods.  Of  these,  taking  first 
the  tie-beam,  we  have  a  piece  subject  to  tension  and  some- 
times to  cross-strain  (see  Art.  682,  Transverse  Strains}.  In 
this  case  the  tensile  strain  "only  need  be  considered.  For 
this  a  rule  is  given  in  Art.  117.  In  this  rule,  if  the  factor  of 
safety  be  taken  at  20,  the  result  will  be  sufficiently  large  to 
allow  for  necessary  cuttings  at  the  joints.  Therefore,  if  the 
beam  be  of  Georgia  pine,  equation  (16.),  Art.  117,  becomes  — 

__  27900x20  _ 
~~     ~ 


or,  say,  35  inches.  This  is  ample  to  resist  the  tensile  strain  ; 
but,  to  resist  the  transverse  strains  to  which  such  a  long 
piece  of  timber  is  subjected  in  the  hands  of  the  workman, 
it  would  be  proper  to  make  it,  say,  6x9. 


STRAIN    UPON   THE   RAFTER.  205 

227. — The  Rafter. — A  rafter,  like  a  post,  is  subject  to  a 
compressive  force,  and  is  liable  to  fail  in  three  ways,  name- 
ly :  by  flexure,  by  being  crushed,  or  by  crushing  the  material 
against  which  it  presses.  To  render  it  entirely  safe,  there- 
fore, it  is  requisite  to  ascertain  the  requirements  for  resisting 
failure  in  each  of  these  three  ways. 

Of  these  it  will  be  convenient  to  consider,  first,  that  of 
the  liability  to  being  crushed.  The  rule  for  this  is  found  in 
Art.  107.  Let  the  rafter  be  of  Georgia  pine,  then  the  vajue 
of  C,  Table  I.,  will  be  9500.  The  strain  in  the  rafter  (Art. 
22$)  is  31,200  pounds.  Now,  taking  the  value  of  a,  the  fac- 
tor of  safety,  at  10,  we  have,  by  Rule  VI.  (Art.  107.) — 

31200  x  10 
A  =      9500       =  32'737: 

or,  33  inches  area  of  cross-section.  This  is  the  size  of  the 
rafter  at  its  smallest  section ;  for  example,  at  any  one  of  the 
joints  where  it  is  customary  to  reduce  the  area  by  cutting 
for  the  struts  and  rods. 

Again :  Let  the  liability  of  the  rafter  to  flexure  be  now 
considered.  For  this  we  have  a  rule  in  Art.  114.  The 
length  of  the  rafter  between  unsupported  points  is  nearly  9$ 
feet,  or  9!  x  12  =  1 16  inches.  Let  the  thickness  of  the  rafter 
be  taken  at  6  inches.  Then,  by  Rule  XI.  (Art.  114),  we 
have — 


+  fyer3)  _  31200  x  10(14-  f  x  -00109  x  r*). 

c  r~  9500  x  6 

/        116 
r=7--g-=i9i;  19*  *    :  373-8- 

Then,  f  x  -00109  x  373-8  =  0-611127 
adding  unity  =  I  • 

1-611127 

Substituting  this,  we  have— 

31200  x  IPX  1-611127  _.  50267 1 '624  _ 
"  9500  x~6~  "57000" 


206  CONSTRUCTION. 

or,  to  resist  flexure  the  breadth  is  required  to  be  8-82,  or, 
say,  9  inches ;  or,  the  rafter  is  to  be  6  xg  inches  at  the  foot. 
The  strain  in  the  rafter  at  the  upper  end  is  only  half  that  at 
the  foot ;  the  area  of  cross-section,  therefore,  at  the  head 
need  not  be  more  than  half  that  which  is  required  at  the 
foot ;  but  it  is  usual  to  make  it  there  about  f  of  the  size  at  the 
foot.  In  this  case  it  would  be,  therefore,  6x6  inches  at  the 
upper  end. 

Lastly,  the  requirement  to  resist  crushing  the  surfaces 
against  which  the  rafter  presses  is  to  be  considered. 

The  fibres  of  timber  yield  much  more  readily  when 
pressed  together  by  a  force  acting  at  right  angles  to  the  di- 
rection of  their  length  than  when  it  acts  in  a  line  with  their 
length. 

The  value  of  timber  subjected  to  pressure  in  these  two 
ways  is  shown  in  Arts.  94,  98.  In  Table  I.,  the  value  per 
square  inch  of  the  first  stated  resistance  is  expressed  by  P, 

and  the  ultimate  resistance  of  the  other  by  — .  The  value 
of  timber  per  square  inch  to  safely  resist  crushing  may  be  ex- 
pressed by  — ,  in  which  a  is  the  factor  of  safety.  Timber 
pressed  in  an  oblique  direction  will  resist  a  force  exceeding 

£ 

that   expressed  by  P,  and  less  than  that  expressed  by  -— . 

ct 

When  the  angle  of  inclination  at  which  the  force  acts  is  just 
45°,  then  the  force  will  be  an  average  between  P  and  — . 

And  for  any  angle  of  inclination,  the  force  will  vary  inverse- 
ly as  the  angle  ;  approaching  P  as  the  angle  is  enlarged, 

but  approaching  -  -  as  the  angle  is  diminished.     It  will  be 

equal  to -- when    the   angle   becomes   zero,   and  equal    P 

when  the  angle  becomes  90°.  The  resistance  of  timber  per 
square  inch  to  an  oblique  force  is  therefore  expressed  by— 


M  = 


RESISTANCE   OF   SURFACES.  2O? 

where  A°  equals  the  complement  of  the  angle  of  inclination. 
In  a  roof,  A°  is  the  acute  angle  formed  by  the  rafter  with 
a  vertical  line.  If  no  convenient  instrument  be  at  hand  to 
measure  the  angle,  describe  an  arc  upon  the  plan  of  the 
truss — thus  :  with  C  B  (Fig.  84)  for  radius,  describe  the  arc 
B  g,  and  get  the  length  of  this  arc  in  feet  by  stepping  it  off 
with  a  pair  of  dividers.  Then — 


where  k  equals  the  length  of  the  arc,  and  h  equals  B  C,  the 
height  of  the  roof.  Therefore — • 

M  = 

equals  the  value  of  timber  per  square  inch  in  a  tie-beam,  C 
and  P  being  obtained  from  Table  I.,  Art.  94.  When  C  for 
the  kind  of  wood  in  the  tie-beam  exceeds  C  set  opposite  the 
kind  of  wood  in  the  rafter,  then  the  latter  is  to  be  used  in 
the  rules  instead  of  the  former. 

The  value  of  M,  equation  (103.),.  is  the  resistance  per 
square  inch  of  the  surface  pressed  at  the  foot  of  the  rafter. 
The  resistance  of  the  entire  surface  will  therefore  be  MA, 
where  A  equals  the  area  of  the  joint.  Then,  when  the  re- 
sistance equals  the  strain,  we  will  have — 


from  which  we  have — 


in  which  5  is  the  strain  to  be  resisted. 

Now,  the  end  of  the  rafter  must  be  of  sufficient  size  to 
afford  a  joint  the  area  of  which  will  not  be  less  than  that 
expressed  by  A  in  equation  (104.). 

For  example,  the  strain  to  which  the  rafter,  Fig.  84,  is 
subject  at  its  foot  is  ascertained  to  be  (Art.  225)  31,200  pounds. 
For  Georgia  pine,  the  material  of  the  tie-beam,  P  =  900 
(Art.  94,  Table  /.),  and  ^  =  9500. 


208  CONSTRUCTION. 

The  length  of  the  arc  Bg  is  about  14-4  feet;  the  height 
B  Cis  13  feet.  Let  a,  the  factor  of  safety,  be  taken  at  10, 
then  we  have  (104.) — 

31200 


900  +  (o-63| 


= 


x  50)          - 

or,  the  superficial  area  of  the  bearing  at  the  joint  required 
to  prevent  crushing  the  tie-beam  is  33^-  inches. 

The  results  of  the  computations  show  that  the  rafter  is 
required  to  be  6  inches  thick,  9  inches  wide  at  the  foot,  and 
6  inches  wide  at  the  top.  It  is  also  ascertained  that,  in  cut- 
ting for  the  bearing  for  the  struts  and  boring  for  the  sus- 
pension-rods, it  is  required  that  there  shall  be  at  least  33 
inches  area  of  cross-section  left  intact  ;  and,  farther,  that  the 
area  of  the  surface  of  the  joint  against  the  tie-beam  should 
not  be  less  than  33^  inches. 

228.  —  The  Braces.  —  Each  brace  is  subject  to  compres- 
sion, and  is  liable  to  fail  if  too  small,  in  the  same  manner  as  the 
rafter.  Its  size  is  to  be  ascertained,  therefore,  in  the  manner 
described  for  the  rafter  ;  which  need  not  be  here  repeated, 
except,  perhaps,  as  to  the  liability  to  fail  by  flexure  ;  for  in  this 
case  we  have  the  breadth  given,  and  need  to  find  the  thick- 
ness. The  breadth  of  the  brace  is  fixed  by  the  thickness  of 
the  rafter,  for  it  is  usual  to  have  the  two  pieces  flush  with 
each  other.  Rule  XI.  (Art.  114)  is  to  be  used,  but  with  this 
difference,  namely  :  instead  of  the  thickness,  use  the  breadth 
as  one  of  the  factors  in  the  divisor.  Thus  — 


(105.) 


In  working  this  rule,  it  is  required,  in  order  to  get  the 
value  of  r,  the  ratio  between  the  height  and  thickness,  to 
assume  the  thickness  before  it  is  ascertained  ;  and  after  com- 
putation, if  the  result  shows  that  the  assumed  value  was  not 
a  near  approximation,  a  second  trial  will  have  to  be  made. 
Usually  the  first  trial  will  be  sufficient. 


STRAIN  UPON  BRACES.  209 

For  example,  the  brace  D  E  is  about  9$  feet  or  1 16  inches 
long.  As  the  strain  in  it  is  only  5200  pounds,  the  thickness 
will  probably  be  not  over  3  inches.  Assuming  it  at  this,  we 

have  r  =  -=  -*-J-&  =  38$  ;  the  square  of  whichxis  about 
Therefore,  we  have-  4  N I  VE  R  S  I  T  Y 

|-XO-OOI09XI495=2 

add  unity  =  i. 

^444S 
The  equation  reduces,  therefore,  to  this — 

,  =  5200  x  10XV4445 

9500  x  6 

or,  the  required  thickness  of  the  brace  is  3^-  inches,  or  the 
brace  should  be,  say,  3 J  x  6  inches.  In  this  case  the  result  is 
so  near  the  assumed  value,  a  second  trial. is  not  needed. 

For  the  second  brace,  we  have  the  length  equal  to  about 
\2\  feet  or  147  inches;  and  the  strain  equal  to  6575  pounds 
(Art.  22$).  The  ratio,  therefore,  may  be  obtained  by  assum- 
ing the  thickness,  say,  at  4.  With  this,  we  have — 


i^—  36-75  ;    the   square  of   which   is    1350^' 
With  this  value  of  r2— 

|  x  « 00109  x  1350^  =  2- 2081 
add  unity  =  I. 

3-2081 
Then — 

6575  x  lox  3-2081  . 

t=.^L2 —     _^ -  =  3.7006. 

9500x6 

Comparing  this  result  with  the  assumed  value  of  /  =  4» 
we  find  the  difference  so  great  as  to  require  a  second  trial. 
As  the  value  of  r  was  taken  too  low,  the  result  obtained  is 
correspondingly  low.  The  true  value  is  somewhere  between 
3  -  7  and  4.  Assume  it  now,  say,  at  3  -  9.  With  this  value,  we 
have — 

r  =  -==  —  =  37-692  ;    the   square   of  which  is    1420-7. 


210  CONSTRUCTION. 

With  this  value  of  r*— 

|  x  -00109  x  1420-7  =  2-32282 
add  unity  =  i  • 

3-32282 
Then  — 

,  _  6575x10x3-32282  _ 
9500x6 

This  result  is  a  trifle  less  than  the  assumed  value,  3-9.  The 
true  value  is  between  these,  and  probably  is  about  3-86. 
This  is  quite  near  enough  for  use.  This  brace,  therefore,  is 
required  to  be  3  -  86  x  6  inches,  or,  say,  4x6  inches. 

229.  —  The  Su§pension-Rods.—  These  are  usually  made  of 
wrought  iron.  This  metal,  when  of  excellent  quality,  may 
be  safely  trusted  with  12,000  pounds  per  inch  sectional  area. 
But  it  is  usual,  for  good  work,  to  compute  the  area  at  only 
9000  pounds  per  inch,  and,  as  ordinarily  made,  these  rods 
ought  not  to  be  loaded  with  more  than  7000  pounds.  The 
strain  divided  by  this  value  per  inch  of  the  metal  will  give 
the  sectional  area  of  cross-section.  For  example,  the  strain  in 
the  rod  D  G,  Fig.  84,  is  867  pounds  (Art.  225);  therefore— 

867 


or,  the  sectional  area  required  is  only  an  eighth  of  an  inch. 
By  reference  to  the  table  of  areas  of  circles  in  the  Appen- 
dix, the  diameter  of  a  rod  containing  the  required  area,  as 
above,  will  be  found  to  be  a  little  less  than  half  an  inch.  A 
rod  half  an  inch  in  diameter  will  therefore  be  of  ample 
strength.  For  appearance's  sake,  however,  no  rod  in  a  truss 
should  be  less  than  f  of  an  inch  in  diameter. 

•The  rod  FE  has  to  resist  a  strain  of  3192  pounds.     For 
this,  then,  we  have  — 


A  reference  to  the  table  of  areas  shows  that  a  rod  contain- 


ROOF-BEAMS.  211 

ing  this  area  would  be  a  little  more  than  J  of  an  inch  in  di- 
ameter ;  it  would  be  of  ample  strength,  say,  at  £  of  an  inch 
in  diameter. 

The  rod  C  B,  at  the  centre,  has  to  carry  a  strain  of  10,167 
pounds.     For  this,  then,-  we  have  — 

10167 


A  reference  to  the  table  of  areas  shows  that  this  rod  should 
be  i£  inches  in  diameter. 

230.  —  Roof-Beams,  Jack-Rafters,  and  Purlins.  —  These 
timbers  are  subject  to  loads  nearly  uniformly  distributed, 
and  their  dimensions  may  be  obtained  by  Rule  XXX.,  equa- 
tion (35.),  Art.  140.  In  this  equation,  U=cfl(Art.  152). 
Substituting  this  value  for  U,  and  r  I  for  tf,  equation  (35.)  be- 
comes — 


and  putting  for  r  the  rate  of  deflection,  .04,  we  have  — 


a  formula  convenient  for  roof-timbers. 

Example.  —  In  a  roof  where  the  roofing  is  to  be  supported 
on  white-pine  roof-beams  10  feet  long,  placed  2\  feet  from 
centres,  and  where  the  load  per  foot  superficial  is  to  be  40 
pounds,  including  wind  and  snow  :  what  should  be  the  di- 
mensions of  the  roof-beams?  By  equation  (106.)— 


bd  = 


Now  if  b,  the  breadth,  be  fixed,  say,  at  3,  then— 


d  =  5-64  nearly. 


212 


CONSTRUCTION. 


The  roof-beams,  therefore,  require  to  be  3  x  5$,  or,  say,  3x6. 
All  pieces  of  timber  subject  to  cross-strains  will  sustain 
safely  much  greater  strains  when  extended  in  one  piece  over 
two,  three,  or  more  distances  between  bearings  ;  therefore, 
roof-beams,  jack-rafters,  and  purlins  should,  if  possible,  be 
made  in  as  long  lengths  as  practicable ;  the  roof-beams  and 
purlins  laid  on,  not  framed  into,  the  principal  rafters,  and 
extended  over  at  least  two  spaces,  the  joints  alternating  on 
the  trusses ;  and  likewise  the  jack-rafters  laid  on  the  purlins 
in  long  lengths. 

231. — Five  Examples  of  Rooft:  are  shown  at  Figs.  85,  86, 
87,  88,  and  89.    In  Fig.  85,  a  is  an  iron  suspension-rod,  by  b  are 

braces.  In  Fig.  86,  a,  a,  and 
b  are  iron  rods,  and  d,  d,  c,  c 
are  braces.  In  Fig.  87,  a,  b 
are  iron  rods,  d,  d  braces,  and 
c  the  straining  beam.  In 
FIG.  85.  Fig.  88,  a,  a,  b,  b  are  iron  rods, 

,  e>  d,  d  are  braces,  and  c  is  a  straining  beam.    In  Fig.  89,  pur- 


out 


S<rft. I 


lins  are  located  at  PP,  etc. ;  the  inclined  beam  that  lies  upon 
them  is  the  jack-rafter;  the  post  at  the  ridge  is  the  king- 


TRUSS   WITH   BUILT-RIB. 


2I3 


rTi 


?k 

o  a 


post,  the  others  are  queen-posts.  In  this  design  the  tie-beam 
is  increased  in  height  along  the  middle  by  a  strengthening 
piece  (Art.  163),  for  the  purpose  of  sustaining  additional 
weight  placed  in  the  room  form- 
ed in  the  truss  (Art.  216). 

Fig.  90  shows  a  method  of 
constructing  a  truss  having  a 
built-rib  in  the  place  of  prin- 
cipal rafters.  The  proper  form 
for  the  curve  is  that  of  the  par- 
abola (Art.  560).  This  curve, 
when  as  flat  as  is  described  in 
the  figure,  approximates  so  close- 
ly to  that  of  the  circle  that  the 
latter  may  be  used  in  its  stead. 
The  height,  a  b,  is  just  half  of 
a  c,  the  curve  to  pass  through 
the  middle  of  the  rib.  The  rib 
is  composed  of  two  series  of 
abutting  pieces,  bolted  together,  oo  ^ 
These  pieces  should  be  as  long 
as  the  dimensions  of  the  timber 
will  admit,  in  order  that  there 
may  be  but  few  joints.  The  sus- 
pending pieces  are  in  halves, 
notched  and  bolted  to  the  tie- 
beam  and  rib,  and  a  purlin  is 
framed  upon  the  upper  end  of 
each.  A  truss  of  this  construc- 
tion needs,  for  ordinary  roofs, 
no  diagonal  braces  between  the 
suspending  pieces,  but  if  extra 
strength  is  required  the  braces 
may  be  added.  The  best  place  \~^~.\ 
for  the  suspending  pieces  is  at 

the  joints  of  the  rib.  A  rib  of  this  kind  will  be  sufficiently 
strong  if  the  area  of  its  section  contain  about  one  fourth 
more  timber  than  is  required  for  that  of  a  rafter  for  a  roof 
of  the  same  size.  The  proportion  of  the  depth  to  the  thick- 
ness should  be  about  as  10  to  7. 


214 


CONSTRUCTION. 


232. — Roof-Tru§s  with  Elevated  Tic-Beam. — Designs 
such  as  are  shown  in  Fig.  91  have  the  tie  elevated  for  the  ac- 
commodation of  an  arch  in  the  ceiling.  This  and  all  similar 
designs  are  seriously  objectionable,  and  should  always  be 


avoided ;  as  the  smail  height  gained  by  the  omission  of  the 
tie-beam  can  never  compensate  for  the  powerful  lateral 
strains  which  are  exerted  by  the  oblique  position  of  the 


supports,  tending  to  separate  the  walls.  Where  an  arch  is 
required  in  the  ceiling,  the  best  plan  is  to  carry  up  the 
walls  as  hi^h  as  the  top  of  the  arch.  Then,  by  using  a 
horizontal  tie-beam,  the  oblique  strains  will  be  entirely  re- 


HIP-ROOFS. 


215 


moved.  It  is  well  known  that  many  a  public  building  has 
been  all  but  ruined  by  the  settling  of  the  roof,  consequent 
upon  a  defective  plan  in  the  formation  of  the  truss  in  this 
respect.  It  is  very  necessary,  therefore,  that  the  horizontal 


FIG.  91. 


tie-beam  be  used,  except  where  the  walls  are  made  so  strong 
and  firm  by  buttresses,  or  other  support,  as  to  prevent  a 
possibility  of  their  separating.  (See  Art.  212.) 


233.— Hip-Roofs:  Lines  and  Bevils.— The  lines  a  b  and 
be  in  Fig.  92,  represent  the  walls  at  the  angle  of  a  building; 
b  e  is  the  seat  of  the  hip-rafter,  and  gfoi  a  jack  or  cnppl( 
rafter.  Draw  e  h  at  right  angles  to  be,  and  make  it  equal 


2l6 


CONSTRUCTION. 


to  the  rise  of  the  roof ;  join  b  and  //,  and  h  b  will  be  the 
length  of  the  hip-rafter.  Through  e  draw  di  at  right  angles 
to  b  c\  upon  b,  with  the  radius  b  h,  describe  the  arc  hi, 
cutting  di  in  z ;  join  b  and  i,  and  extend  gfto  meet  b  i  in/; 
then  gj  will  be  the  length  of  the  jack-rafter.  The  length 
of  each  jack-rafter  is  found  in  the  same  manner — by  extend- 
ing its  seat  to  cut  the  line  b  i.  From  /  draw  fk  at  right 
angles  tofg,  also//  at  right  angles  to  be\  make/£  equal 
to//  by  the  arc  Ik,  or  make  g  k  equal  to  gj  by  the  arc  jk ; 
then  the  angle  at  /  will  be  the  top-bevil  of  the  jack-rafters, 
and  the  one  at  k  will  be  the  down-bevil* 

234. — The  Backing  of  the  Hip-Rafter. — At  any  con- 
venient place  in  be  (Fig.  92),  as  o,  draw  m  n  at  right  angles  to 
b  e ;  from  o,  tangical  to  b  h,  describe  a  semicircle,  cutting  b  e 
in  s ;  join  m  and  s  and  n  and  s\  then  these  lines  will  form  at 
s  the  proper  angle  for  bevilling  the  top  of  the  hip-rafter. 

DOMES.f 

235. — Domes. — The  usual  form  for  domes  is  that  of  the 
sphere ;  the  base  circular.  When  the  interior  dome  does  not 


FIG.  93. 

rise  too  high,  a  horizontal  tie  may  be  thrown  across,  by 
which  any  degree  of  strength  required  may  be  obtained. 


*  The  lengths  and  bevils  of  rafters  for  root-valleys  can  also  be  found  by  the 
above  process. 

f  See  also  Art.  68. 


CONSTRUCTION   OF  DOMES. 


217 


-  93  shows  a  section,  and  Fig.  94  the  plan,  of  a  dome  of 
this  kind,  a  b  being  the  tie-beam  in  both.  Two  trusses  of 
this  kind  (Fig.  93),  parallel  to  each  other,  are  to  be  placed 
one  on  each  side  of  the  opening  in  the  top  of  the  dome. 
Upon  these  the  whole  framework  is  to  depend  for  support, 


FIG.  94. 


u    u 


and  their  strength  must  be  calculated  accordingly.  (See 
Arts.  70  to  80  and  214  to  222.)  If  the  dome  is  large  and  of 
importance,  two  other  trusses  may  be  introduced  at  right 
angles  to  the  foregoing,  the  tie-beams  being  preserved  in 


;i*/   FIG.  95. 

one  continuous  length  by  framing  them  high  enough  to  pass 
over  the  others. 

236.— Ribbed  Dome.— When  the  interior  must  be  kept 
free,  then  the  framing  may  be  composed  of  a  succession  of 
ribs  standing  upon  a  continuous  circular  curb  of  timber,  as 


218 


CONSTRUCTION. 


seen  at  Figs.  95  and  96 — the  latter  being  a  plan  and  the  former 
a  section.  This  curb  must  be  well  secured,  as  it  serves  hi 
the  place  of  a  tie-beam  to  resist  the  lateral  thrust  of  the  ribs. 
In  small  domes  these  ribs  may  be  easily  cut  from  wide 
plank ;  but  where  an  extensive  structure  is  required,  they 
must  be  built  in  two  thicknesses  so  as  to  break  joints,  in  the 
same  manner  as  is  described  for  a  roof  at  Art.  231.  They 
should  be  placed  at  about  two  feet  apart  at  the  base,  and 
strutted  as  at  a  in  Fig.  95. 


FIG.  .96. 

The  scantling  of  each  thickness  of  the  rib  may  be  as  fol- 
lows : 

For  domes  of  24  feet  diameter,  i    x    8  inches. 
36      "  "  i£  x  10       " 

2      x  13          " 


a 

"     60 
"     90 

"108 


3    x  13 


237. — Dome:  Curve  of  Equilibrium. — The  surfaces  of  a 
dome  may  be  finished  to  any  curve  that  may  be  desired,  but 
the  framing  should  be  constructed  of  such  form  that  the 
curve  of  equilibrium  shall  be  sure  to  pass  through  the  middle 
of  the  depth  of  the  framing.  The  nature  of  this  curve  is 
such  that,  if  an  arch  or  dome  be  constructed  in  accordance 
with  it,  no  one  part  of  the  structure  will  be  less  capable  than 
another  of  resisting  the  strains  and  pressures  to  which  the 
whole  fabric  may  be  exposed.  The  curve  of  equilibrium  for 
an  arched  vault  or  a  roof,  where  the  load  is  equally  diffused 


CURVE   OF   EQUILIBRIUM. 


2I9 


over  the  whole  surface,  is  that  of  a.  parabola  (Art.  460}-,  for 
a  dome  having  no  lantern,  tower,  or  cupola  above  it,  a  cubic 
parabola  (Fig.  97) ;  and  for  one  having  a  tower,  etc.,  above  it, 
a  curve  approaching  that  of  an  hyperbola  must  be  adopted, 
as  the  greatest  strength  is  required  at  its  upper  parts.  If 
the  curve  of  a  dome  be  circular  (as  in  the  vertical  section, 
^•95)> the  pressure  will  have  a  tendency  to  burst  the  dome 
outwards  at  about  one  third  of  its  height.  Therefore,  when 
this  form  is  used  in  the  construction  of  an  extensive  dome, 
an  iron  band  should  be  placed  around  the  framework  at 
that  height ;  and  whatever  may  be  the  form  of  the  curve,  a 
band  or  tie  of  some  kind  is  necessary  around  or  across  the 
base. 


0 

o/ 

0 

/ 

/I 

1 

of  \ 

A 

» 


s  f 


FIG.  97. 


If  the  framing  be  of  a  form  less  convex  than  the  curve  of 
equilibrium,  the  weight  will  have  a  tendency  to  crush  the 
ribs  inwards,  but  this  pressure  may  be  effectually  overcome 
by  strutting  between  the  ribs ;  and  hence  it  is  important 
that  the  struts  be  so  placed  as  to  form  continuous  horizontal 
circles. 

238. — Cubic  Parabola  Computed. — Let  a  b  (Fig.  97)  be 
the  base,  and  b  c  the  height.  Bisect  a  b  at  d,  and  divide  a  d 
into  100  equal  parts  ;  of  these  give  d e  26,  ef  \%\,fg  i&,gh 
\2\,  h  i  lof,  ij  9^,  and  the  balance,  8j,  to/# ;  divide  be  into 
8  equal  parts,  and  from  the  points  of  division  draw  lines 
parallel  to  a  b,  to  meet  perpendiculars  from  the  several  points 


22O 


CONSTRUCTION. 


of  division  in  a  b,  at  the  points  o,  o,  oy  etc.     Then  a  curve 
traced  through  these  points  will  be  the  one  required. 

239. — Small  Domes  over  Stairways :  are  frequently  made 
elliptical  in  both  plan  and  section  ;  and  as  no  two  of  the  ribs 


in  one  quarter  of  the  dome  are  alike  in  form,  a  method  for 
obtaining  the  curves  may  be  useful. 


FIG.  99, 

To  find  the  curves  for  the  ribs  of  an  elliptical  dome,  let 
abed  (Fig.  98)  be  the  plan  of  a  dome,  and  ef  the  seat  of 
one  of  the  ribs,  Then  take  c /for  the  transverse  axis  and 
twice  the  rise,  og>  of  the  dome  for  the  conjugate,  and  de- 


COVERING   OF   DOMES. 


221 


scribe  (according  to  Arts.  548,  549,  etc.)  the  semi-ellipse 
e gf,  which  will  be  the  curve  required  for  the  rib  e gf.  The 
other  ribs  are  found  in  the  same  manner. 

240. — Covering  for  a  Spherical  Dome. — To  find  the 
shape,  let^4  C^T-99)  be  the  plan,  and  B  the  section,  of  a  given 
dome.  From  a  draw  a  c  at  right  angles  to  a  b  ;  find  the 
stretch-out  (Art.  524)  of  o  b,  and  make  dc  equal  to  it;  divide 
the  arc  o  b  and  the  line  d  c  each  into  a  like  number  of  equal 
parts,  as  5  (a  large  number  will  insure  greater  accuracy  than 
a  small  one)  ;  upon  c,  through  the  several  points  of  division 
in  cd,  describe  the  arcs  o  do,  i  e  I,  2/2,  etc. ;  make  do  equal 
to  half  the  width  of  one  of  the  boards,  and  draw  o  s  parallel 


to  a  c ;  join  s  and  #,  and  from  the  points  of  division  in  the  arc 
<?£drop  perpendiculars,  meeting  a  sinij  kl\  from  these 
points  draw  z  4,/3,  etc.,  parallel  to  ac ;  make  dotei,  etc.,  on 
the  lower  side  of  a  c,  equal  to  do,e\,  etc.,  on  the  upper  side  ; 
trace  a  curve  through  the  points  o,  1,2,  3,  4,  c,  on  each  side 
of  dc ;  then  o  c  o  will  be  the  proper  shape  for  the  board.  By 
dividing  the  circumference  of  the  base  A  into  equal  parts, 
and  making  the  bottom,  o  d  o,  of  the  board  of  a  size  equal  to 
one  of  those  parts,  every  board  may  be  made  of  the  same 
size.  In  the  same  manner  as  the  above,  the  shape  of  the 
covering  for  sections  of  another  form  may  be  found,  such  as 
an  ogee,  cove,  etc. 

To  find  the  curve  of  the  boards  when  laid  in  horizontal 
courses,  let  A  B  C  (Fig.  100)  be  the  section  of  a  given  dome, 


222 


CONSTRUCTION. 


and  DB  its  axis.  Divide  B  C  into  as  many  parts  as  there 
are  to  be  courses  of  boards,  in  the  points  i,  2,  3,  etc. ;  through 
i  and  2  draw  a  line  to  meet  the  axis  extended  at  a\  then  a 
will  be  the  centre  for  describing  the  edges  of  the  board  F. 
Through  3  and  2  draw  3  b;  then  b  will  be  the  centre  for  de- 


FlG.    IOI. 

scribing  F.  Through  4  and  3  draw  4^;  then  d  will  be  the 
centre  for  G.  B  is  the  centre  for  the  arc  i  o.  If  this  method 
is  taken  to  find  the  centres  for  the  boards  at  the  base  of  the 
dome,  they  would  occur  so  distant  as  to  make  it  impracti- 
cable ;  the  following  method  is  preferable  for  this  purpose  : 
G  being  the  last  board  obtained  by  the  above  method,  ex- 
tend the  curve  of  its  inner  edge  until  it  meets  the  axis,  D  B, 


in  e ;  from  3,  through  e,  draw  3  f,  meeting  the  arc  A  B  in  f ; 
join  f  and  4,  f  and  5 ,  and  /and  6,  cutting  the  axis,  D  B,  in  s,  n, 
and  m  ;  from  4,  5,  and  6  draw  lines  parallel  to  A  C  and  cutting 
the  axis  in  c,  p,  and  r ;  make  c  4  (Fig.  101)  equal  to  c  4  in  the  pre- 
vious figure,  and  c  s  equal  to  c  s  also  in  the  previous  figure  ; 
then  describe  the  inner  edge  of  the  board  //,  according  to 
Art.  516;  the  outer  edge  can  be  obtained  by  gauging  from 
the  inner  edge.  Tn  like  manner  proceed  to  obtain  the  next 


DESIGNS    OF   BRIDGES. 


223 


board — taking/  5  for  half  the  chord,  and  p  n  for  the  height 
of  the  segment.  Should  the  segment  be  too  large  to  be  de- 
scribed easily,  reduce  it  by  finding  intermediate  points  in  the 
curve,  as  at  Art.  515. 

241. — Polygonal  Dome:  Form  of  Angle-Rib. — To  ob- 
tain the  shape  of  this  rib,  let  A  G  H  (Fig.  102)  be  the  plan  of 
a  given  dome,  and  C  D  a  vertical  section  taken  at  the  line 
ef.  From  i,  2,  3,  etc.,  in  the  arc  CD  draw  ordinates,  paral- 
lel to  A  D,  to  meet  f-G ;  from  the  points  of  intersection  on 
fG  draw  ordinates  at  right  angles  to/ G\  make  s  i  equal 
to  o  i,  s  2  equal  to  o  2,  etc. ;  then  GfB,  obtained  in  this  way, 
will  be  the  angle-rib  required.  The  best  position  for  the 
sheathing-boards  for  a  dome  of  this  kind  is  horizontal,  but  if 
they  are  required  to  be  bent  from  the  base  to  the  vertex, 
their  shape  may  be  found  in  a  similar  manner  to  that  shown 
at  Fig.  99. 

BRIDGES. 


FIG.  103 

242.— Bridges.— Of  plans  for  the  construction  of  bridges, 
perhaps  the  following  are  the  most  useful.  Fig.  103  shows  a 
method  of  constructing  wooden  bridges  where  the  banks 
of  the  river  are  high  enough  to  permit  the  use  of  the  tie- 
beam,  a  b.  The  upright  pieces,  c  d,  are  notched  and  bolted 
on  in  pairs,  for  the  support  of  the  tie-beam.  A  bridge  ot 
this  construction  exerts  no  lateral  pressure  upon  the  abut- 
ments. This  method  may  be  employed  even  where  the  banks 
of  the  river  are  low,  by  letting  the  timbers  for  the  roadway 
rest  immediately  upon  the  tie-beam.  In  this  case  the  irame- 
work  above  will  serve  the  purpose  of  a  railing. 


224 


CONSTRUCTION. 


Fig.  104  exhibits  a  wooden  bridge  without  a  tie-beam. 
Where  staunch  buttresses  can  be  obtained  this  method  may 
be  recommended  ;  but  if  there  is  any  doubt  ot  their  stability, 
it  should  not  be  attempted,  as  it  is  evident  that  such  a  sys- 
tem of  framing  is  capable  of  a  tremendous  lateral  thrust. 


FIG.  104. 

243, — Bridge§:  Built-Rib. — Fig.  105  represents  a  bridge 
with  a  built-rib  (see  Art.  231)  as  a  chief  support.  The  curve 
of  equilibrium  will  not  differ  much  from  that  of  a  parabola ; 
this,  therefore,  may  be  used — especially  if  the  rib  is  made 


FIG.  105. 

gradually  a  little  stronger  as  it  approaches  the  buttresses. 
As  it  is  desirable  that  a  bridge  be  kept  low,  the  following 
table  is  given  to  show  the  least  rise  that  may  be  given  to  the 
rib. 


Span  in  Feet. 

Least  Rise  in  Feet 

Span  in  Feet. 

Least  Rise  in  Feet 

Span  in  Feet.  !  Least  Rise  in  Feet 

( 

30 

o-5 

1  2O 

7 

280 

24 

40 

0-8 

I4O 

8 

3OO 

28 

50 

1-4 

1  60 

10 

320 

32 

60 

2 

1  80 

ii 

350 

39 

70 

21 

200 

12 

380 

47 

80 

3 

2  2O 

14 

400 

53 

90 

4 

240 

17 

100 

5 

260 

20 

i 

W 
((UNIVERSITY 

DIMENSIONS  OF  THE  BUILT-Rl] 


The  rise  should  never  be  made  less  than  this,  but  in  all 
cases  greater  if  practicable  ;  as  a  small  rise  requires  a  greater 
quantity  of  timber  to  make  the  bridge  equally  strong.  The 
greatest  uniform  weight  with  which  a  bridge  is  likely  to  be 
loaded  is,  probably,  that  of  a  dense  crowd  of  people.  This 
may  be  estimated  at  70  pounds  per  square  foot,  and  the  fram- 
ing and  gravelled  roadway  at  230  pounds  more ;  which 
amounts  to  300  pounds  on  a  square  foot.  The  following 
rule,  based  upon  this  estimate,  may  be  useful  in  determining 
the  area  of  the  ribs. 

Rule  LXVIL— Multiply  the  width  of  the  bridge  by  the 
square  of  half  the  span,  both  in  feet,  and  divide  this  pro- 
duct by  the  rise  in  feet  multiplied  by  the  number  of  ribs  ; 
the  quotient  multiplied  by  the  decimal  o-oon  will  give  the 
area  of  each  rib  in  feet.  When  the  roadway  is  only  planked, 
use  the  decimal  0-0007  instead  of  o-ooii, 

Example. — What  should  be  the  area  of  the  ribs  for  a 
bridge  of  200  feet  span,  to  rise  15  feet  and  be  30  feet  wide, 
with  three  curved  ribs  ?  The  half  of  the  span  is  100,  and 
its  square  is  10000  ;  this  multiplied  by  30  gives  300000,  and 
15  multiplied  by  3  gives  45;  then  300000  divided  by  45 
gives  6666|,  which  multiplied  by  o-ooii  gives  7-333  feet  or 
1056  inches  for  the  area  of  each  rib.  Such  a  rib  may  be  24 
inches  thick  by  44  inches  deep,  and  composed  of  6  pieces, 
2  in  width  and  3  in  depth. 

The  above  rule  gives  the  area  of  a  rib  that  would  be 
requisite  to  support  the  greatest  possible  tiniform  load. 
But  in  large  bridges,  a  variable  load,  such  as  a  heavy  wagon, 
is  capable  of  exerting  much  greater  strains ;  in  such  cases, 
therefore,  the  rib  should  be  made  larger.* 

In  constructing  these  ribs,  if  the  span  be  not  over  50  feet, 
each  rib  may  be  made  in  two  or  three  thicknesses  of  timber 
(three  thicknesses  is  preferable),  of  convenient  lengths  bolted 
together ;  but  in  larger  spans,  where  the  rib  will  be  such  as 
to  render  it  difficult  to  procure  timber  of  sufficient  breadth, 
they  may  be  constructed  by  bending  the  pieces  to  the  proper 
curve  and  bolting  them  together.  In  this  case,  where  tim- 


*  See  Tredgold's  Carpentry  by  Hurst,  Arts.  174  to  177. 


226 


CONSTRUCTION. 


her  of  sufficient  length  to  span  the  opening-  cannot  be  ob- 
tained, and  scarfing  is  necessary,  such  joints  must  be  made 
as  will  resist  both  tension  and  compression  (see  Fig.  1 14). 
To  ascertain  the  greatest  depth  for  the  pieces  which  compose 
the  rib,  so  that  the  process  of  bending  may  not  injure  their 
elasticity,  multiply  the  radius  of  curvature  in  feet  by  the 
decimal  0-05,  and  the  product  will  be  the  depth  in  inches. 
Example. — Suppose  the  curve  of  the  rib  to  be  described 
with  a  radius  of  100  feet,  then  what  should  be  the  depth  ?  The 
radius  in  feet,  100,  multiplied  by  0-05  gives  a  product  of  5 
inches.  White  pine  or  oak  timber  5  inches  thick  would 
freely  bend  to  the  above  curve  ;  and  if  the  required  depth 
of  such  a  rib  be  20  inches,  it  would  have  to  be  composed  of  at 
least  4  pieces.  Pitch  pine  is  not  quite  so  elastic  as  white 
pine  or  oak — its  thickness  may  be  found  by  using  the  deci- 
mal 0-046  instead  of  0-05. 


FIG.  106. 


244. — Bridges:  Framed  Rib. — In  spans  of  over  250 
feet,  SL  framed  r\by  as  in  Fig.  106,  would  be  preferable  to  the 
foregoing.  Of  this,  the  upper  and  the  lower  edges  are 
formed  as  just  described,  by  bending  the  timber  to  the  proper 
curve.  The  pieces  that  tend  to  the  centre  of  the  curve, 
called  radials,  are  notched  and  bolted  on  in  pairs,  and  the 
cross-braces  are  halved  together  in  the  middle,  and  abut  end 
to  end  between  the  radials.  The  distance  between  the  ribs 
of  a  bridge  should  not  exceed  about  8  feet.  The  roadway 
should  be  supported  by  vertical  standards  bolted  to  the  ribs 


THE    ROADWAY  AND   ABUTMENTS.  22/ 

at  about  every  10  to  15  feet.  At  the  place  where  they  rest 
on  the  ribs,  a  double,  horizontal  tie  should  be  notched  and 
bolted  on  the  back  of  the  ribs,  and  also  another  on  the  un- 
derside ;  and  diagonal  braces  should  be  framed  between  the 
standards,  over  the  space  between  the  ribs,  to  prevent  lat- 
eral motion.  The  timbers  for  the  roadway  may  be  as  light 
as  their  situation  will  admit,  as  all  useless  timber  is  only  an 
unnecessary  load  upon  the  arch. 

245. — Bridges:  Roadway. — If  a  roadway  be  18  feet 
wide,  two  carriages  can  pass  without  inconvenience.  Its 
width,  therefore,  should  be  either  9,  18,  27,  or  36  feet,  ac- 
cording to  the  amount  of  travel.  The  width  of  the  foot- 
path should  be  two  feet  for  every  person.  When  a  stream 
of  water  has  a  rapid  current,  as  few  piers  as  practicable 
should  be  allowed  to  obstruct  its  course ;  otherwise  the 
bridge  will  be  liable  to  be  swept  away  by  freshets.  When 
the  span  is  not  over  300  feet,  and  the  banks  of  the  river  are 
of  sufficient  height  to  admit  of  it,  only  one  arch  should  be 
employed.  The  rise  of  the  arch  is  limited  by  the  form  of 
the  roadway,  and  by  the  height  of  the  banks  of  the  river 
(see  Art.  243).  The  rise  of  the  roadway  should  not  exceed 
one  in  24  feet,  but  as  the  framing  settles  about  one  in  72,  the 
roadway  should  be  framed  to  rise  one  in  18,  that  it  may  be 
one  in  24  after  settling.  The  commencement  of  the  arch  at 
the  abutments — the  spring,  as  it  is  termed — should  not  be 
below  high-water  mark  ;  and  the  bridge  should  be  placed  at 
right  angles  with  the  course  of  the  current. 

246. — Bridges:  Abutments. — The  best  material  for  the 
abutments  and  piers  of  a  bridge  is  stone ;  and  no  other 
should  be  used.  The  following  rule  is  to  determine  the  ex- 
tent of  the  abutments,  they  being  rectangular,  and  built  with 
stone  weighing  120  pounds  to  a  cubic  foot. 

Rule  LXVIII  —  Multiply  the  square  of  the  height  of  the 
abutment  by  160,  and  divide  this  product  by  the  weight  of  a 
square  foot  of  the  arch,  and  by  the  rise  of  the  arch  ;  add 
unity  to  the  quotient,  and  extract  the  square  root.  Dimin- 
ish the  square  root  by  unity,  and  multiply  the  root  so  dimin- 


228  CONSTRUCTION. 

ished  by  half  the  span  o-f  the  arch,  and  by  the  weight  of  a 
square  foot  of  the  arch.  Divide  the  last  product  by  120 
times  the  height  of  the  abutment,  and  the  quotient  will  be 
the  thickness  of  the  abutment. 

Example. — Let  the  height  of  the  abutment  from  the  base 
to  the  springing  of  the  arch  be  20  feet,  hall  the  span  100  feet, 
the  weight  of  a  square  foot  of  the  arch,  including  the  great- 
est possible  load  upon  it,  300  pounds,  and  the  rise  of  the  arch 
18  feet:  what  should  be  its  thickness?  The  square  of  the 
height  of  the  abutment,  400,  multiplied  by  160  gives  64000, 
and  300  by  18  gives  5400;  64000  divided  by  5400  gives  a 
quotient  of  11-852;  one  added  to  this  makes  12-852,  the 
square  root  of  which  is  3-6  ;  this,  less  one  is  2-6  ;  this  mul- 
tiplied by  100  gives  260,  and  this  again  by  300  gives  78000; 
this  divided  by  120  times  the  height  of  the  abutment,  2400, 
gives  32  feet  6  inches,  the  thickness  required. 

The  dimensions  of  a  pier  will  be  found  by  the  same  rule  ; 
for,  although  the  thrust  of  an  arch  may  be  balanced  by  an 
adjoining  arch  when  the  bridge  is  finished,  and  while  it  re- 
mains uninjured,  yet,  during  the  erection,  and  in  the  event 
of  one  arch  being  destroyed,  the  pier  should  be  capable  of 
sustaining  the  entire  thrust  of  the  other. 

Piers  are  sometimes  constructed  of  timber  their  princi- 
pal strength  depending  on  piles  driven  into  the  earth ;  but 
such  piers  should  never  be  adopted  where  it  is  possible  to 
avoid  them  ;  for,  being  alternately  wet  and  dry,  they  decay 
much  sooner  than  the  upper  parts  of  the  bridge.  Spruce 
and  elm  are  considered  good  for  piles.  Where  the  height 
from  the  bottom  of  the  river  to  the  roadway  is  great,  it  is  a 
good  plan  to  cut  them  off  at  a  little  below  low-water  mark, 
cap  them  with  a  horizontal  tie,  and  upon  this  erect  the  posts 
for  the  support  of  the  roadway.  This  method  cuts  off  the 
pan  that  is  continually  wet  from  that  which  is  only  occa- 
sionally so,  and  thus  affords  an  opportunity  for  replacing  the 
upper  part.  The  pieces  which  are  immersed  will  last  a 
great  length  of  time,  especially  when  of  elm  ;  for  it  is  a 
well-established  fact  that  timber  is  less  durable  when  subject 
to  alternate  dryness  and  moisture  than  when  it  is  either  con- 
tinually wet  or  continually  dry.  It  has  been  ascertained  that 


CENTRING   FOR   BRIDGES. 


229 


the  piles  under  London  Bridge,  after  having  been  driven 
about  600  years,  were  not  materially  decayed.  These  piles 
are  chiefly  of  elm,  and  wholly  immersed. 

247. — Centre§  for  Stone  Bridges. — Fig.  107  is  a  design 
for  a  centre  for  a  stone  bridge  where  intermediate  supports, 
as  piles  driven  into  the  bed  of  the  river,  are  practicable.  Its 
timbers  are  so  distributed  as  to  sustain  the  weight  of  the 
arch-stones  as  they  are  being  laid,  without  destroying  the 
original  form  of  the  centre  ;  and  also  to  prevent  its  destruc- 
tion or  settlement,  should  any  of  the  piles  be  swept  away. 
The  most  usual  error  in  badly-constructed  centres  is  that 
the  timbers  are  disposed  so  as  to  cause  the  framing  to 
rise  at  the  crown  during  the  laying  of  the  arch-stones  up 


FIG.  107. 

the  sides.  To  remedy  this  evil,  some  have  loaded  the  crown 
with  heavy  stones ;  but  a  centre  properly  constructed  will 
need  no  such  precaution. 

Experiments  have  shown  that  an  arch-stone  does  not  press 
upon  the  centring  until  its  bed  is  inclined  to  the  horizon  at 
an  angle  of  from  30  to  45  degrees,  according  to  the  hardness 
of  the  stone,  and  whether  it  is  laid  in  mortar  or  not.  For 
general  purposes,  the  point  at  which  the  pressure  com- 
mences may  be  considered  to  be  at  that  joint  which  forms 
an  angle  of  32  degrees  with  the  horizon.  At  this  point  the 
pressure  is  inconsiderable,  but  gradually  increases  towards 
the  crown.  The  following  table  gives  the  portion  of  the 
weight  of  the  arch-stones  that  presses  upon  the  framing  at 
the  various  angles  of  inclination  formed  by  the  bed  of  the 


230  CONSTRUCTION. 

stone  with  the  horizon.  The  pressure  perpendicular  to  the 
curve  is  equal  to  the  weight  of  the  arch-stone  multiplied  by 
the  decimal — 

•o,  when  the  angle  of  inclination  is  32  degrees. 
.04  "  "  "  34 

•  08  "  "  "  36         " 

•  12  "  "  "  38  " 
•17                  "                    "                   "                    40  " 
•21                   "                    "                   "                   42            " 

-25  "  -  »  44 

•  29  "  "  "  46  " 

•  33  "  ;"  "  48  " 

•  37  "  "  "  50  " 

•  4  "  "  "  52  " 

•44  "  4t  "  54  " 

•48  "  "  "  56 

•  52  "  "  u  58  " 
-54  "  "  "  60  " 

From  this  it  is  seen  that  at  the  inclination  of  44  degrees  the 
pressure  equals  one  quarter  the  weight  of  the  stone  ;  at  57 
degrees,  half  the  weight ;  and  when  a  vertical  line,  as  a  b 
(Fig.  1 08),  passing  through  the  centre  of 
gravity  of  the  arch-stone,  does  not  fall 
within  its  bed,  c  d,  the  pressure  may  be  con- 
sidered equal  to  the  whole  weight  of  the 
stone.  This  will  be  the  case  at  about  60 
degrees,  when  the  depth  of  the  stone  is 
double  its  breadth.  The  direction  of  these 
pressures  is  considered  in  a  line  with  the  radius  of  the  curve. 
The  weight  upon  a  centre  being  known,  the  pressure  may  be 
estimated  and  the  timber  calculated  accordingly.  But  it 
must  be  remembered  that  the  whole  weight  is  never  placed 
upon  the  framing  at  once — as  seems  to  have  been  the  idea 
had  in  view  by  the  designers  of  some  centres.  In  building 
the  arch,  it  should  be  commenced  at  each  buttress  at  the 
same  time  (as  is  generally  the  case),  and  each  side  should 
progress  equally  towards  the  crown.  In  designing  the  fram- 


CENTRE   FOR  A  STONE   BRIDGE. 


231 


ing,  the  effect  produced  by  each  successive  layer  of  stone 
should  be  considered.  The  pressure  of  the  stones  upon  one 
side  should,  by  the  arrangement  of  the  struts,  be  counter- 
poised by  that  of  the  stones  upon  the  other  side. 

Over  a  river  whose  stream  is  rapid,  or  where  it  is  neces- 
sary to  preserve  an  uninterrupted  passage  for  the  purposes 
of  navigation,  the  centre  must  be  constructed  without  in- 
termediate supports,  and  without  a  continued  horizontal  tie 
at  the  base  ;  such  a  centre  is  shown  at  Fig.  109.  In  laying 
the  stones  from  the  base  up  to  a  and  c,  the  pieces  bd  and 
bd  act  as  ties  to  prevent  any  rising  at  b.  After  this,  while 
the  stones  are  being  laid  from  a  and  from  c  to  b,  they  act  as 
struts;  the  piece  f  g  is  added  for  additional  security. 
Upon  this  plan,  with  some  variation  to  suit  circumstances, 


FIG.  109. 

centres  may  be  constructed  for  any  span  usual  in  stone- 
bridge  building. 

In  bridge  centres,  the  principal  timbers  should  abut,  and 
not  be  intercepted  by  a  suspension  or  radial  piece  between. 
These  should  be  in  halves,  notched  on  each  side  and  bolted. 
The  timbers  should  intersect  as  little  as  possible,  for  the 
more  joints  the  greater  is  the  settling ;  and  halving  them 
together  is  a  bad  practice,  as  it  destroys  nearly  one  half  the 
strength  of  the  timber.  Ties  should  be  introduced  across, 
especially  where  many  timbers 'meet ;  and  as  the  centre  is 
to  serve  but  a  temporary  purpose,  the  whole  should  be  de- 
signed with  a  view  to  employ  the  timber  afterwards  for 
other  uses.  For  this  reason,  all  unnecessary  cutting  should 
be  avoided. 


232  CONSTRUCTION. 

Centres  should  be  sufficiently  strong-  to  preserve  a 
staunch  and  steady  form  during  the  whole  process  of  build- 
ing ;  for  any  shaking  or  trembling  will  have  a  tendency  to 
prevent  the  mortar  or  cement  from  setting.  For  this  pur- 
pose, also,  the  centre  should  be  lowered  a  trifle  immedi- 
ately after  the  key-stone  is  laid,  in  order  that  the  stones  may 
take  their  bearing  before  the  mortar  is  set ;  otherwise  the 
joints  will  open  on  the  underside.  The  trusses,  in  centring, 
are  placed  at  the  distance  of  from  4  to  6  feet  apart,  accord- 
ing to  their  strength  and  the  weight  of  the  arch.  Between 
every  two  trusses  diagonal  braces  should  be  introduced  to 
prevent  lateral  motion. 

In  order  that  the  centre  may  be  easily  lowered,  the 
frames,  or  trusses,  should  be  placed  upon  wedge-formed 
sills,  as  is  shown  at  d  (Fig.  109).  These  are  contrived  so  as 
to  admit  of  the  settling  of  the  frame  by  driving  the  wedge 
d  with  a  maul,  or,  in  large  centres,  with  a  piece  of  timber 
mounted  as  a  battering-ram.  The  operation  of  lowering  a 
centre  should  be  very  slowly  performed,  in  order  that  the 
parts  of  the  arch  may  take  their  bearing  uniformly.  The 
wedge  pieces,  instead  of  being  placed  parallel  with  the 
truss,  are  sometimes  made  sufficiently  long  and  laid  through 
the  arch,  in  a  direction  at  right  angles  to  that  shown  at  Fig. 
109.  This  method  obviates  the  necessity  of  stationing  men 
beneath  the  arch  during  the  process  of  lowering ;  and  was 
originally  adopted  with  success  soon  after  the  occurrence  of 
an  accident,  in  lowering  a  centre,  by  which  nine  men  were 
killed. 

To  give  some  idea  of  the  manner  of  estimating  the  pres- 
sures, in  order  to  select  timber  of  the  proper  scantling,  cal- 
culate the  pressure  (Art.  247)  of  the  arch-stones  from  i  to  b 
(Fig.  109),  and  suppose  half  this  pressure  concentrated  at  a, 
and  acting  in  the  direction  a  f.  Then,  by  the  parallelogram 
of  forces  (Art.  71),  the  strain  in  the  several  pieces  compos- 
ing the  frame  bda  may  be  computed.  Again,  calculate 
the  pressure  of  that  portion  of  the  arch  included  between  a 
and  c,  and  consider  half  of  it  collected  at  £,  and  acting  in  a 
vertical  direction  ;  then,  by  the  parallelogram  of  forces,  the 
pressure  on  the  beams  bd  and  £</may  be  found.  Add  the 


JOINTS   OF   THE   ARCH-STONES. 


233 


pressure  of  that  portion  of  the  arch  which  is  included  be- 
tween i  and  b  to  half  the  weight  of  the  centre,  and  consider 
this  amount  concentrated  at  d,  and  acting  in  a  vertical  direc- 
tion ;  then,  by  constructing  the  parallelogram  of  forces,  the 
pressure  upon  dj  may  be  ascertained. 

The  strains  having  been  obtained,  the  dimensions  of  the 
several  pieces  in  the  frames  £#</and  bed  may  be  found  by 
computation,  as  directed  in  the  case  of  roof-trusses,  from 
Arts.  226  to  229.  The  tie-beams  b  d,  b  d,  if  made  of  suffi- 
cient size  to  resist  the  compressive  strain  acting  upon  them 
from  the  load  at  b,  will  be  more  than  large  enough  to  resist 
the  tensile  strain  upon  them  during  the  laying  of  the  first 
part  of  the  arch-stones  below  a  and  c. 

248.— Arcli-Stones  :  JToint§.— In  an  arch,  the  arch-stones 
are  so  shaped  that  the  joints  between  them  are  perpendicu- 
lar to  the  curve  of  the  arch,  or  to  its  tangent  at  the  point  at 
which  the  joint  intersects  the  curve.  In  a  circular  arch,  the 


FIG.  no. 


joints  tend  toward  the  centre  of  the  circle  ;  in  an  elliptical 
arch,  the  joints  may  be  found  by  the  following  process : 
To  find  the  direction  of  the  joints  for  an  elliptical  arch ; 


FIG.  in. 


a  joint  being  wanted  at  a  (Fig.  no),  draw  lines  from  that 
point  to  the  foci, /and/;  bisect  the  angle  faf  with  the 
line  a  b  ;  then  a  b  will  be  the  direction  of  the  joint. 


234  CONSTRUCTION. 

To  find  the  direction  of  the  joints  for  a  parabolic  arch ; 
a  joint  being-  wanted  at  a  (Fig.  1 1 1),  draw  a  e  at  right  angles 
to  the  axis  eg;  make  eg  equal  to  c e,  and  join  a  and  g\ 
draw  a  h  at  right  angles  to  ag\  then  a  h  will  be  the 
direction  of  the  joint.  The  direction  of  the  joint  from  b  is 
found  in  the  same  manner.  The  lines  a  gaud  b  f  are  tan- 
gents to  the  curve  at  those  points  respectively  ;  and  any 
number  of  joints  in  the  curve  may  be  obtained  by  first 
ascertaining  the  tangents,  and  then  drawing  lines  at  right 
angles  to  them.  (See  Art.  462.) 

JOINTS. 

24-9. — Timber  Joints. — The  joint  shown  in  Fig.  112  is 
simple  and  strong ;  but  the  strength  consists  wholly  in  the 
bolts,  and  in  the  friction  of  the  parts  produced  by  screwing 
the  pieces  firmly  together.  Should  the  timber  shrink  to 


FIG.  ii2. 


even  a  small  degree,  the  strength  would  depend  altogether 
on  the  bolts.  It  would  be  made  much  stronger  by  indent- 
ing the  pieces  together,  as  at  the  upper  edge  of  the  tie-beam 


FIG.  113. 


in  Fig.  113,  or  by  placing  keys  in  the  joints,  as  at  the  lower 
edge  in  the  same  figure.  This  process,  however,  weakens 
the  beam  in  proportion  to  the  depth  of  the  indents. 


FIG.  114 


Fig.  114  shows  a  method  of  scarfing,  or  splicing,  a  tie- 
beam  without  bolts.  The  keys  are  to  be  of  well-seasoned, 
hard  wood,  and,  if  possible,  very  cross-grained.  The  addi- 


THE   SPLICING  OF   TIMBER.  235. 

tion  of  bolts  would  make  this  a  very  strong  splice,  or  even 
white-oak  pins  would  add  materially  to  its  strength. 

Fig.  1 1 5  shows  about  as  strong  a  splice,  perhaps,  as  can 
well  be  made.  It  is  to  be  recommended  for  its  simplicity  ; 
as,  on  account  of  there  being  no  oblique  joints  in  it,  it  can 
be  readily  and  accurately  executed.  A  complicated  joint  is 
the  worst  that  can  be  adopted ;  still,  some  have  proposed 
joints  that  seem  to  have  little  else  besides  complication  to 
recommend  them. 

In  proportioning  the  parts  of  these  scarfs,  the  depths  of 


FIG.  115. 

all  the  indents  taken  together  should  be  equal  to  one  third 
of  the  depth  of  the  beam.  In  oak,  ash  or  elm,  the  whole 
length  of  the  scarf  should  be  six  times  the  depth,  or  thick- 
ness, of  the  beam,  when  there  are  no  bolts  ;  but,  if  bolts  in- 
stead of  indents  are  used,  then  three  times  the  breadth  ;  and 
when  both  methods  are  combined,  twice  the  depth  of  the 
beam.  The  length  of  the  scarf  in  pine  and  similar  soft 
woods,  depending  wholly  on  indents,  should  be  about  12 
times  the  thickness,  or  depth,  of  the  beam  ;  when  depend- 
ing wholly  on  bolts,  6  times  the  breadth ;  and  when  both 
methods  are  combined,  4  times  the  depth. 


FIG.  116. 

Sometimes  beams  have  to  be  pieced  that  are  required 
to  resist  cross-strains — such  as  a  girder,  or  the  tie-beam  of  a 
roof  when  supporting  the  ceiling.  In  such  beams,  the 
fibres  of  the  wood  in  the  upper  part  are  compressed  ;  and 
therefore  a  simple  butt  joint  at  that  place  (as  in  Fig.  116) 
is  far  preferable  to  any  other.  In  such  case,  an  oblique 
joint  is  the  very  worst.  The  under  side  of  the  beam  being 
in  a  state  of  tension,  it  must  be  indented  or  bolted,  or  both  ; 
and  an  iron  plate  under  the  heads  of  the  bolts  gives  a  great 
addition  of  strength. 


236  CONSTRUCTION. 

Scarfing  requires  accuracy  and  care,  as  all  the  indents 
should  bear  equally ;  otherwise,  one  being  strained  more 
than  another,  there  would  be  a  tendency  to  splinter  off  the 
parts.  Hence  the  simplest  form  that  will  attain  the  object 
is  by  far  the  best.  In  all  beams  that  are  compressed  end- 
wise, abutting  joints,  formed  at  right  angles  to  the  direction 
of  their  length,  are  at  once  the  simplest  and  the  best.  For  a 
temporary  purpose,  Fig.  112  would  do  very  well ;  it  would 
be  improved,  however,  by  having  a  piece  bolted  on  all  four 
sides.  Fig.  113,  and  indeed  each  of  the  others,  since  they 
have  no  oblique  joints,  would  resist  compression  well. 

In  framing  one  beam  into  another  for  bearing  purposes, 
such  as  a  floor-beam  into  a  trimmer,  the  best  place  to  make 
the  mortise  in  the  trimmer  is  in  the  neutral  line  (Arts.  120, 
121),  which  is  in  the  middle  of  its  depth.  Some  have 
thought  that,  as  the  fibres  of  the  upper  edge  are  compressed, 


FIG.  117. 

a  mortise  might  be  made  there,  and  the  tenon  driven  in 
tight  enough  to  make  the  parts  as  capable  of  resisting  the 
compression  as  they  would  be  without  it ;  and  they  have 
therefore  concluded  that  plan  to  be  the  best.  This  could 
not  be  the  case,  even  if  the  tenon  would  not  shrink  ;  for  a 
joint  between  two  pieces  cannot  possibly  be  made  to  resist 
compression  so  well  as  a  solid  piece  without  joints.  The 
proper  place,  therefore,  for  the  mortise  is  at  the  middle  of 
the  depth  of  the  beam  ;  but  the  best  place  for  the  tenon,  in 
the  floor-beam,  is  at  its  bottom  edge.  For  the  nearer  this 
is  placed  to  the  upper  edge,  the  greater  is  the  liability  for  it 
to  splinter  off;  if  the  joint  is  formed,  therefore,  as  at  Fig.  \  17, 
it  will  combine  all  the  advantages  that  can  be  obtained.  Dou- 
ble tenons  are  objectionable,  because  the  piece  framed  into 
is  needlessly  weakened,  and  the  tenons  are  seldom  so  accu- 
rately made  as  to  bear  equally.  For  this  reason,  unless  the  tusk 


THE   FRAMING   IN  A   ROOF-TRUSS. 


237 


at  a  in  the  figure  fits  exactly,  so  as  to  bear  equally  with  the 
tenon,  it  had  better  be  omitted.  And  in  sawing  the  shoulders 
care  should  be  taken  not  to  saw  into  the  tenon  in  the  least, 
as  it  would  wound  the  beam  in  the  place  least  able  to  bear  it. 

Thus  it  will  be  seen  that  framing  weakens  both  pieces, 
more  or  less.  It  should,  therefore,  be  avoided  as  much  as 
possible  ,  and  where  it  is  practicable  one  piece  should  rest 
upon  the  other,  rather  than  be  framed  into  it.  This  re- 
mark applies  to  the  bearing  of  floor-beams  on  a  girder,  to 
the  purlins  and  jack-rafters  of  a  roof,  etc. 

In  a  framed  truss  for  a  roof,  bridge,  partition,  etc.,  the 
joints  should  be  so  constructed  as  to  direct  the  pressures 
through  the  axes  of  the  several  pieces,  and  also  to  avoid 
every  tendency  of  the  parts  to  slide.  To  attain  this  object, 


FIG.  118. 


FIG.  119. 


FIG.  120. 


the  abutting  surface  on  the  end  of  a  strut  should  be  at 
right  angles  to  the  direction  of  the  pressure  ;  as  at  the  joint 
shown  in  Fig.  1 18  for  the  foot  of  a  rafter  (see  Art.  86),  in  Fig. 
1 19  for  the  head  of  a  rafter,  and  in  Fig.  120  for  the  foot  of  a 
strut  or  brace.  The  joint  at  Fig.  118  is  not  cut  completely 
across  the  tie-beam,  but  a  narrow  lip  is  left  standing  in  the 
middle,  and  a  corresponding  indent  is  made  in  the  rafter,  to 
prevent  the  parts  from  separating  sideways.  The  abutting 
surface  should  be  made  as  large  as  the  attainment  of  other 
necessary  objects  will  admit.  The  iron  strap  is  added  to 
prevent  the  rafter  sliding  out,  should  the  end  of  the  tie- 
beam,  by  decay  or  otherwise,  splinter  off.  In  making-  the 
joint  shown  at  Fig.  119,  it  should  be  left  a  little  open  at  a, 
so  as  to  bring  the  parts  to  a  fair  bearing  at  the  settling  of 
the  truss,  which  must  necessarily  take  place  from  the  shrink- 
ing of  the  king-post  and  other  parts.  If  the  joint  is  made 
fair  at  first,  when  the  truss  settles  it  will  cause  it  to  open  at 


CONSTRUCTION. 


the  under  side  of  the  rafter,  thus  throwing  the  whole  pres- 
sure upon  the  sharp  edge  at  a.  This  will  cause  an  indenta- 
tion in  the  king-post,  by  which  the  truss  will  be  made  to 
settle  further ;  and  this  pressure  not  being  in  the  axis  of  the 
rafter,  it  will  be  greatly  increased,  thereby  rendering  the 
rafter  liable  to  split  and  break. 


FIG.  121. 


FIG.  122. 


FIG.  123. 


If  the  rafters  and  struts  were  made  to  abut  end  to  end, 
as  in  Figs.  121,  122  and  123,  and' the  king  or  queen  post 
notched  on  in  halves  -and  bolted,  the  ill  effects  of  shrinking 
would  be  avoided.  This  method  has  been  practised  with 
success  in  some  of  the  most  celebrated  bridges  and  roofs  in 
Europe  ;  and,  were  its  use  adopted  in  this  country,  the  un- 
seemly sight  of  a  hogged  ridge  would  seldom  be  met  with. 


FIG.  124. 


FIG.  125. 


A  plate  of  cast-iron   between  the   abutting   surfaces   will 
equalize  the  pressure. 

Fig.  124  is  a  proper  joint  for  a  collar-beam  in  a  small 
roof :  the  principle  shown  here  should  characterize  all  tie- 
joints.  The  dovetail  joint,  although  extensively  practised 
in  the  above  and  similar  cases,  is  the  very  worst  that  can  be 
employed.  The  shrinking  of  the  timber,  if  only  to  a  small 


WHITE-OAK    PINS   AND   IRON   STRAPS.  239 

degree,  permits  the  tie  to  withdraw — as  is  shown  at  Fig. 
12$.  The  dotted  line  shows  the  position  of  the  tie  after  it 
has  shrunk. 

Locust  and  white-oak  pins  are  great  additions  to  the 
strength  of  a  joint.  In  many  cases  they  would  supply  the 
place  of  iron  bolts  ;  and,  on  account  of  their  small  cost,  they 
should  be  used  in  preference  wherever  the  strength  ef  iron 
is  not  requisite.  In  small  framing,  good  cut  nails  are  of 
great  service  at  the  joints  ;  but  they  should  not  be  trusted 
to  bear  any  considerable  pressure,  as  they  are  apt  to  be 
brittle.  Iron  straps  are  seldom  necessary,  as  all  the  joinings 
in  carpentry  may  be  made  without  them.  They  can  be 
used  to  advantage,  however,  at  the  foot  of  suspending-pieces, 
and  for  the  rafter  at  the  end  of  the  tie-beam.  In  roofs  for 
ordinary  purposes,  the  iron  straps  for  suspending-pieces 
may  be  as  follows :  When  the  longest  unsupported  part  of 
the  tie-beam  is — 

10  feet,  the  strap  may' be  i  inch  wide  by  T3T  thick. 

15     "  "  "  i£     "       •    "       i      " 

2Q       «  «  «  2          u  «  JL          " 

In  fastening  a  strap,  its  hold  on  the  suspending-piece  will  be 
much  increased  by  turning  its  ends  into  the  wood.  Iron 
straps  should  be  protected  from  rust ;  for  thin  plates  of  iron 
decay  very  soon,  especially  when  exposed  to  dampness. 
For  this  purpose,  as  soon  as  the  strap  is  made  let  it  be 
heated  to  about  a  blue  heat,  and,  while  it  is  hot,  pour  over 
its  entire  surface  raw  linseed  oil,  or  rub  it  with  beeswax. 
Either  of  these  will  give  it  a  coating  which  dampness  will 
not  penetrate. 


SECTION  III.— STAIRS. 

250. — §tair§  :  General  Requirements. — The  STAIRS  is 
that  commodious  arrangement  of  steps  in  a  building  by 
which  access  is  obtained  from  one  story  to  another.  Their 
position,  form,  and  finish,  when  determined  with  discrimi- 
nating taste,  add  greatly  to  the  comfort  and  elegance  of  a 
structure.  As  regards  their  position,  the  first  object  should 
be  to  have  them  near  the  middle  of  the  building,  in  order 
that  they  may  afford  an  equally  easy  access  to  all  the  rooms 
and  passages.  Next  in  importance  is  light ;  to  obtain  which 
they  would  seem  to  be  best  situated  near  an  outer  wall,  in 
which  windows  might  be  constructed  for  the  purpose  ;  yet 
a  skylight,  or  opening  in  the  roof,  would  not  only  provide 
light,  and  so  secure  a  central  position  for  the  stairs,  but  may 
be  made,  also,  to  assist  materially  as  an  ornament  to  the 
building,  and,  what  is  of  more  importance,  afford  an  oppor- 
tunity for  better  ventilation. 

All  stairs,  especially  those  of  the  most  important  build- 
ings, should  be  erected  of  stone  or  some  equally  durable  and 
fire-resisting  material,  that  the  means  of  egress  from  a  burn- 
ing building  may  not  be  too  rapidly  destroyed. 

Winding  stairs,  or  those  in  which  the  direction  is  gradu- 
ally changed  by  means  of  winders,  or  steps  which  taper  in 
width,  are  interesting  by  reason  of  the  greater  skill  required 
in  their  construction  ;  but  are  objectionable,  for  the  reason 
that  children  are  exposed  to  accident  by  their  liability  to  fall 
when  passing  over  the  narrow  ends  of  the  steps.  Stairs  of 
this  kind  should  be  tolerated  only  where  there  is  not  suffi- 
cient space  for  those  with  flyers,  or  steps  of  parallel  width. 

Stairs  in  one  long  continuous  flight  are  also  objection- 
able. Platforms  or  landings  should  be  introduced  at  inter- 
vals, so  that  any  one  flight  may  not  contain  more  than  about 
twelve  or  fifteen  steps. 

The  width  of  stairs  should  be  in  accordance  with  the  im- 


KHORSABAD.— ASSYRIAN   TEMPLE,    RESTORED. 


THE  GRADE   OF   STAIRS.  241 

portance  of  the  building  in  which  they  are  placed,  varying 
from  3  to  12  feet.  Where  two  persons  are  expected  to  pass 
each  other  conveniently  the  least  width  admissible  is  3  feet. 
Still,  in  crowded  cities,  where  land  is  valuable,  the  space 
allowed  for  passages  is  correspondingly  small,  and  in  these 
stairs  are  sometimes  made  as  narrow  as  2\  feet. 

From  3  to  4  feet  is  a  suitable  width  for  a  good  dwelling  ; 
while  5  feet  will  be  found  ample  for  stairs  in  buildings  occu- 
pied by  many  people ;  and  from  8  to  12  feet  is  sufficient  for 
the  width  of  stairs  in  halls  of  assembly. 

To  avoid  tripping  or  stumbling,  care  should  be  exer- 
cised, in  the  planning  of  a  stairs,  to  secure  an  even  grade. 
To  this  end,  the  nosing,  or  outer  edge,  of  each  step  should  be 
exactly  in  line  with  all  the  other  nosings.  In  stairs  com- 
posed of  both  flyers  and  winders,  precaution  in  this  regard 
is  especially  needed.  In  such  stairs,  the  steps — flyers  and 
winders  alike — should  be  of  one  width  on  the  line  along 
which  a  person  would  naturally  walk  when  having  his  hand 
upon  the  rail.  This  tread-line,  consequently,  would  be  paral- 
lel with  the  hand-rail,  and  is  usually  taken  at  a  distance  of 
from  1 8  to  20  inches  from  the  centre  of  it.  In  the  plan  of 
the  stairs  this  tread-line  should  be  drawn  and  divided  into 
equal  parts,  each  part  being  the  tread,  or  width  of  a  flyer 
from  the  face  of  one  riser  to  the  face  of  the  next. 

251. — The  Grade  of  Stairs. — The  extra  exertion  required 
in  ascending  a  staircase  over  that  for  walking  on  level 
ground  is  due  to  the  weight  which  a  person  at  each  step  is 
required  to  lift ;  that  is,  the  weight  of  his  own  body.  Hence 
the  difficulty  of  ascent  will  be  in  proportion  to  the  height  of 
each  step,  or  to  the  rise,  as  it  is  termed.  To  facilitate  the 
operation  of  going  up  stairs,  therefore,  the  -risers should  be 
low.  The  grade  of  a  stairs,  or  its  angle  of  ascent,  depends 
not  only  upon  the  height  of  the  riser,  but  also  upon  the 
width  of  the  step  ;  and  this  has  a  certain  relation  to  the 
riser ;  for  the  width  of  a  step  should  be  in  proportion  to  the 
smallness  of  the  angle  of  ascent. 

The  distance  from  the  top  of  one  riser  to  the  top  of  the 
next  is  the  distance  travelled  at  each  step  taken,  and  this  dis- 


242  STAIRS. 

tance  should  vary  as  the  grade  of  the  stairs ;  for  a  person 
who  in  climbing  a  ladder,  or  a  nearly  vertical  stairs,  can 
travel  only  12  inches,  or  less,  at  a  step,  will  be  able  with 
equal  or  greater  facility  to  travel  at  least  twice  this  distance 
on  level  ground.  The  distance  travelled,  therefore,  should 
be  in  proportion  inversely  to  the  angle  of  ascent ;  or,  the  di- 
mensions of  riser  and  step  should  be  reciprocal :  a  low  rise 
should  have  a  wide  step,  and  a  high  rise  a  narrow  step. 

252. — Pitch-Board :  Relation  of  Rise  to  Tread. —  Among 
the  various  devices  for  determining  the  relation  of  the  rise 
to  the  tread,  or  net  width  of  step,  one  is  to  make  the  sum  of 
the  two  equal  to  18  inches. 

For  example,  for  a  rise  of  6  inches  the  tread  should  be 
12,  for  7  inches  the  tread  should  be  n  ;  or — 

6  +  12    =  18  8    +  10  =  18 
6J  +  iii=  18  8i  +   9i  =  18 

7  +  ii    =  18  9    +  9    =  18 
;£+  ioi=  18  Qj  +    8i=  18 

This  rule  is  simple,  but  the  results  in  extreme  cases  are  not 
satisfactory.  If  the  ascent  of  a  stairs  be  gradual  and  easy, 
the  length  from  the  top  of  one  rise  to  that  of  another,  or  the 
hypothenuse  of  the  pitch-board,  may  be  proportionally  long ; 
but  if  the  stairs  be  steep,  the  length  must  be  shorter. 

There  is  a  French  method,  introduced  by  Blondel  in  his 
Cours  d'  Architecture.  It  is  referred  to  in  Gwilt's  Encyclo- 
pedia, Art.  2813.  • 

This  method  is  based  upon  the  assumed  distance  of  24 
inches  as  being  a  convenient  step  upon  level  ground,  and 
upon  12  inches  as  the  most  convenient  height  to  rise  when 
the  ascent  is  vertical.  These  are  French  inches,  old  system. 
The  24  inches  French  equals  about  25^  inches  English. 

With  these  distances  as  base  and  perpendicular,  a  right- 
angled  triangle  is  formed,  which  is  used  as  a  scale  upon 
which  the  proportions  of  a  pitch-board  are  found.  For 
example,  let  a  line  be  drawn  from  any  point  in  the  hypothe- 
nuse of  this  triangle  to  the  right  angle  of  the  triangle  ;  then 
this  line  will  equal  the  length  of  the  pitch-board,  along  the 


PROPORTIONS   OF   THE   PITCH-BOARD.  243 

rake,  for  a  stairs  having  a  grade  equal  to  the  angle  formed 
by  this  line  and  the  base-line  of  the  scale. 

In  the  absence  of  the  triangular  scale,  the  lengths  of  the 
pitch-boards,  as  found  by  this  rule,  may  be  computed  by  this 
expression— 

W=2$3r-2A;  (107.) 

in  which  W equals  the  tread,  or  base  of  the  pitch-board,  and 
h  the  riser,  or  its  perpendicular  height. 
For  example,  let  //  =  6  ;  then — 

^=25^-2x6=13^. 

This  result  is  greater  than  would  be  proper  in  some  cases. 

The  length  of  the  hypothenuse  of  the  pitch-board  should 
be  proportional  not  only  to  the  angle  of  ascent  (Art.  251),  but 
also  to  the  strength  and  height  of  the  class  of  people  who 
are  to  use  the  stairs.  Tall  and  strong  persons  will  take 
longer  steps  than  short  and  feeble  people.  The  hypothe- 
nuse of  the  pitch-board  should  be  made  in  proportion  to 
the  distance  taken  at  a  step  on  level  ground  by  the  persons 
who  are  to  use  the  stairs. 

If  people  are  divided  into  two  classes,  one  composed  of 
robust  workmen  and  the  other  of  delicate  women  and  in- 
firm men,  then  there  may  be  two  scales  formed  for  the  pitch- 
boards  of  stairs — one  to  be  used  for  shops  and  factories,  and 
the  other  for  dwellings.  The  distance  on  level  ground  trav- 
elled per  step,  by  men,  varies  from  about  26  to  32  inches,  or 
on  an  average  28  inches.  The  height  to  which  men  are 
accustomed  to  rise  on  ladders  is  from  12  to  16  inches  at  each 
step,  or  on  the  average  14  inches. 

With  these  dimensions,  therefore,  of  14  and  28  inches,  a 
scale  may  be  formed  for  pitch-boards  for  stairs,  in  buildings 
to  be  used  exclusively  by  robust  workmen.  And  with  12 
and  24  inches  another  scale  may  be  formed  for  pitch-boards 
for  stairs,  in  buildings  to  be  used  by  women  and  feeble 
people.  These  two  scales  are  both  shown  in  Fig.  126. 
They  are  made  thus  :  Let  C  A  B  be  a  right  angle.  Make  A 
B  equal  to  28  inches,  and  A  C  equal  to  14 ;  then  join  B  and 


244 


STAIRS. 


C.  At  right  angles  to  C  B,  from  A,  draw  A  F\  then  with 
A  F  for  radius  describe  the  arc  F  G.  Then  a'  line,  as  A  K 
or  A  L,  drawn  from  A  at  any  angle  with  A  B  and  limited  by 
the  line  G  FB  will  give  the  length  of  the  hypothenuse  of 
the  pitch-board,  for  shop  stairs  of  a  grade  equal  to  the  angle 
which  said  line  makes  with  A  B.  From  K,  perpendicular  to 
A  B,  draw  K  N ";  then  K  N  will  be  the  proper  riser  for  a 
pitch-board  of  which  A  N  is  the  tread.  So,  likewise,  L  M 
will  be  the  appropriate  riser  for  the  tread  A  M.  The  arc  F  G 
is  introduced  to  limit  the  rake-line  of  pitch-boards  occur- 
ring between  F  and  C,  in  order  to  avoid  making  them  longer 
than  the  one  at  F.  The  scale  for  the  stairs  for  dwellings  is 
made  in  the  same  manner ;  A  D  —  24  inches  being  the  base, 
A  E  =  12  inches  the  rise,  and  J  H  D  the  line  limiting  the 
rake-lines  of  pitch-boards. 


M 


N 

FIG.  126. 


To  compute  the  length  of  risers  and  treads,  we  have  for 
the  scale  for  shops,  for  those  occurring  between  F  and  B — 

r  =  4(28-/):  (108.) 

fi=28-2rj  (109.) 

and  for  those  between  F  and  G,  we  have — 

(108,  A.) 
(109,  A.) 


r= 


-/2; 


1/756^8- 


For  the  scale  for  dwellings,  we  have,  for  those  occurring 
between  H  and  D — 

r  =i(24-/);  (108,  B.) 

t  =  24  —  2  r  ;  (109,  B.) 


STAIRS   FOR   SHOPS   AND   FOR   DWELLINGS. 

and  for  those  between  H  and  J,  we  have — 


t  = 


245 


,  c.) 

(109,  C.) 


where,  in  each  equation,  r  represents  the  riser,  and  t  the 
tread,  or  net  step. 

By  these  formulae,  the  following  tables  have  been  com- 
puted : 

STAIRS  FOR  SHOPS. 


Rise. 

Tread. 

Ratio—  Rise  to  Tread. 

Rise. 

Tread. 

Ratio—  Rise  to  Tread. 

2- 

24- 

to    12- 

7-40 

13-20 

to       -78 

3' 

22- 

"     7-33 

7-60 

12-80 

"       -68 

3-50 

21- 

"     6- 

7-80 

12-40 

"       -59 

4- 

20- 

"     5- 

8- 

12- 

•50 

4-50 

19. 

"      4-22 

8-20 

II-6 

"       -41 

5' 

18- 

"     3'6o 

8-50 

II- 

•29 

5-4 

17-20 

3-19 

8-80 

10-40 

"       -18 

5-7 

16-60 

'     2-91 

9' 

10- 

"       -ii 

6- 

16- 

'     2-67 

9-30 

9-40 

"       -01 

6-25 

I5-50 

:     2-48 

9-60 

8-80 

"    0-92 

6-50 

15- 

'     2-31 

10- 

8- 

.    '     0-80 

6-70 

14-60 

'     2-18 

10-50 

7' 

4    0-67 

6-90 

14-20 

'    2-06 

ii- 

6- 

1     °'55 

7- 

14- 

'       2- 

11-50 

4-95 

'     0-43 

7-20 

13-60 

'       I-89 

12- 

3-58 

i  '     0-30 

STAIRS  FOR  DWELLINGS. 


Rise. 

Tread. 

Ratio—  Rise  to  Tread. 

Rise. 

Tread. 

Ratio  —  Rise  to  Tread. 

2- 

2O- 

I     to    10- 

7-40 

9-20 

i  .to       -24 

3" 

18- 

I      "      6- 

7-50 

9' 

I      "         -2O 

3-50 

17- 

i      '     4-86 

7'60 

8-80 

"         .16 

4' 

16- 

I      '     4- 

7-70 

8-60 

"          -12 

4-50 

15- 

'     3-33 

7-80 

8-40 

"          -08 

5' 

14- 

2-80 

7-90 

8-20 

"       -04 

5-40 

13-20 

'     2-44 

8- 

8- 

• 

5-70 

12-60 

"      2-21 

8-10 

7-80 

"    0-96 

6- 

12- 

"      2- 

8-30 

7-40 

"    0-89 

6-25 

II-5O 

"          -84 

8-50 

7- 

"    0-82 

6-50 

II- 

"          -69 

8-75 

6-50 

"     o-74 

6-75 

IO-5O 

'          -56 

9' 

6- 

"    0-67 

7- 

10- 

•43 

9-30 

5-40 

"    0-58 

7-10 

9-80 

'       -38 

9-60 

4-80 

"    0-50 

7-20 

9-60 

•33 

10- 

3.90 

"    0-39 

7-30 

9.40 

-29 

10-50 

2  -2O 

'      0-21 

These  tables  will  be  useful  in  determining  questions  in- 


246  STAIRS. 

volving  the   proportion  between  the   rise  and  tread   of   a 
pitch-board. 

For  stairs  in  which  the  run  is  limited,  to  determine  the 
number  of  risers  which  would  give  an  easy  ascent :  Divide 
the  run  by  the  height,  and  find  in  the  proper  table,  above, 
the  ratio  nearest  to  the  quotient,  and  in  a  line  with  this  ratio, 
in  the  second  column  to  the  left,  will  be  found  the  corre- 
sponding riser.  With  this  divide  the  rise  in  inches  ;  the  quo- 
tient, or  the  nearest  whole  number  thereto,  will  be  the  required 
number  of  risers  in  the  stairs. 

Example. — For  the  stairs  in  a  dwelling,  let  the  rise  be  12' 
8",  or  I524nches.  Let  the  run  between  the  extreme  risers 
be  17'  2".  To  this,  for  the  purpose  of  obtaining  the  correct 
angle  of  ascent,  by  having  an  equal  number  of  risers  and 
treads,  add,  for  one  more  tread,  say  10  inches,  its  probable 
width;  thus  making  the  total  run  1 8  feet,  or  216  inches. 
Thus  we  have  for  the  run  216,  and  for  the  rise  152.  Divid- 
ing the  former  by  the  latter  gives  i  -42  nearly.  In  the  table 
of  stairs  for  dwellings,  the  ratio  nearest  to  this  is  I  -43,  and  in 
the  line  to  the  left,  in  the  second  column,  is  7,  the  approxi- 
mate size  of  riser  appropriate  to  this  case.  Dividing  the 
rise,  152  inches,  by  this  7,  we  have  2 if  as  the  quotient. 

This  is  nearer  to  22  than  to  21  ;  therefore,  the  number  of 
risers  required  is  22. 

When  the  number  of  risers  is  determined,  then  the  rise 
divided  by  this  number  will  give  the  height  of  each  riser ; 
thus,  in  the  above  case,  the  rise  is  152  inches.  This  divided 
by  22  gives  6-909  inches  for  the  height  of  the  riser. 

When  the  height  of  the  riser  is  known,  then,  if  the  run  is 
unlimited,  the  width  of  tread  will  be  found  in  the  proper  table 
above.  For  example,  if  the  riser  is  7  inches  or  nearly  that, 
then  in  the  table  of  stairs  for  dwellings,  in  the  next  column 
to  the  right,  and  opposite  7  in  the  column  of  risers,  is  found 
10,  the  approximate  width  of  tread.  By  the  use  of  equation 
(109,  B.),  the  width  may  be  had  exactly  according  to  the 
scale.  For  example,  equation  (109,  B.)  with  6-91  for  the 
riser,  becomes— 

t  —  24  —  2  x  6-91  =  io- 18, 
or  about  ioT3^  inches. 


TO   CONSTRUCT   THE    PITCH-BOARD.  247 

When  the  run  is  limited  and  the  number  of  risers  is 
known,  then  the  width  of  tread  is  obtained  by  dividing  the 
run  by  the  number  of  treads.  There  are  always  of  treads 
one  less  than  there  are  of  risers,  in  each  flight. 


253. — Dimensions  of  the  Pitch-Board. — The  first  thing 
in  commencing  to  build  a  stairs  is  to  make  the  //fc/j-board ; 
this  is  done  in  the  following  manner :  Obtain  very  accurate- 
ly, in  feet  and  inches,  the  rise,  or  perpendicular  height,  of  the 
story  in  which  the  stairs  are  to  be  placed.  This  must  be 
taken  from  the  top  of  the  lower  floor  to  the  top  of  the  upper 
floor.  Then,  to  obtain  the  number  of  rises  and  treads  and 
their  size,  proceed  as  directed  in  Art.  252.  Having  obtained 
these,  the  pitch-board  may  be  made  in  the  following  man- 
ner: Upon  a  piece  of  well-seasoned  board  about  -|  of  an 
inch  thick,  having  one  edge  jointed 
straight  and  square,  lay  the  corner 
of  a  steel  square,  as  shown  at  Fig.  127. 
Make  a  b  equal  to  the  riser,  and  b  c 
equal  to  the  tread ;  mark  along  the 
edges  with  a  knife,  and  cut  by  the  FlG<  I27' 

marks,  making  the  edges  of  the  pitch  -  board  perfectly 
square.  The  grain  of  the  wood  should  run  in  the  direction 
indicated  in  the  figure,  because,  in  case  of  shrinkage,  the 
rise  and  the  tread  will  be  equally  affected  by  it.  When  a 
pitch-board  is  first  made,  the  dimensions  of  the  riser  and 
tread  should  be  preserved  in  figures,  in  order  that,  in  case 
of  shrinkage  or  damage  otherwise,  a  second  may  be 
made. 


254. — The  String  of  a  Stairs.  — The  space  required  for 
timber  and  plastering  under  the  steps  is  about  5  inches  for 
ordinary  stairs,  or  6  inches  if  furred ;  set  a  gauge,  there- 
fore, at  5  or  6  inches,  as  the  case  requires,  and  run  it  on  the 
lower  edge  of  the  plank,  as  ab  (Fig.  128).  Commencing  at 
one  end,  lay  the  longest  side  of  the  pitch-board  against  the 
gauge-mark,  a  b,  as  at  c,  and  draw  by  the  edges  the  lines  for 
the  first  rise  and  tread ;  then  place  it  successively  as  at  d,  e, 


248 


STAIRS. 


and  f,  until  the  required  number  of  risers  shall  be  laid  down. 
To  insure  accuracy,  it  is  well  to  ascertain  the  theoretical 
raking  length  of  the  pitch-board  by  computation,  as  in  note 
to  Art.  536,  by  getting  the  square  root  of  the  sum  of  the 
squares  of  the  rise  and  run,  and  using  this  by  which  to 
divide  the  line  ab  into  equal  parts. 


f 


FIG.  128. 


255. — Step  and  Riser  Connection. — Fig.  129  represents 
a  section  of  step  and  riser,  joined  after  the  most  approved 
method.  In  this,  a  represents  the  end  of  a  block  about  2 


FIG.  129, 

inches  long,  two  or  three  of  which,  in  the  length  of  the 
step,  are  glued  in  the  corner.  The  cove  at  b  is  planed  up 
square,  glued  in,  and  stuck  or  moulded  after  the  glue  is 
set. 


PLATFORM    STAIRS. 


256. — Platform  Stair*  :  the  Cylinder. — A  platform  stairs 
ascends  from  one  story  to  another  in  two  or  more  flights, 
having  platforms  or  landings  between  for  resting  and  to 
change  their  direction.  This  kind  of  stairs,  being  simple,  is 


CYLINDER   OF   PLATFORM   STAIRS. 


249 


easily  constructed,  and  at  the  same  time  is  to  be  preferred 

to  those  with  winders,  for  the  convenience  it  affords  in  use 

(Art.  250).     The  cylinder  may  be  of 

any  diameter  desirable,  from  a  few 

inches  to  3  or  more    feet,  but  it  is 

generally  small,  about  6  inches.     It 

may   be   worked    out  of   one    solid 

piece,  but  a  better  way  is  to  glue 

together  3  pieces,  as  in  Fig.  130 ;  in 

which  the  pieces  a,  b,  and  c  compose 

the  cylinder,  and  d  and  e  represent 

parts  of  the  strings.     The  strings, 

after  being  glued  to  the  cylinder,  are  secured  with  screws. 

The  joining  at  o  and  o  is  the  most  proper  for  that  kind  of 

joint. 


FIG.  130. 


FIG.  131. 


257.  — Form  of  Lower  Edge  of  Cylinder.  —  Find  the 

stretch-out,  de  (Fig.  131),  of  the  face  of  the  cylinder,  a  be, 


2?O  STAIRS. 

according  to  Art.  524;  from  d  and  e  draw  df  and  eg  at 
right  angles  to  d^;  draw  kg  parallel  to  de,  and  make  hf 
and  £-2  each  equal  to  one  riser;  from  i  and  f  draw  ij  and 
/£  parallel  to  hg\  place  the  tread  of  the  pitch-board  at  these 
last  lines,  and  draw  by  the  lower  edge  the  lines  kh  and  il ; 
parallel  to  these  draw  m  n  and  op,  at  the  requisite  distance 
for  the  dimensions  of  the  string ;  from  s,  the  centre  of  the 
plan,  draw  sq  parallel  to  df\  divide  //  q  and  qg  each  into  two 
equal  parts,  as  at  v  and  w  ;  from  v  and  w  draw  v  n  and  w  o 
parallel  to  fd\  join  n  and  o,  cutting  qs  in  r ;  then  the  angles 
unr  and  rot,  being  eased  off  according  to  Art.  521,  will  give 
the  proper  curve  for  the  bottom  edge  of  the  cylinder.  A 
centre  may  be  found  upon  which  to  describe  these  curves, 
thus  :  from  u  draw  u  x  at  right  angles  to  m  n  ;  from  r  draw 
r x  at  right  angles  to  no  ;  then  x  will  be  the  centre  for  the 
curve  ur.  The  centre  for  the  curve  rt  may  be  found  in  a 
similar  manner.  Centres  from  which  to  strike  these  curves 
are  usually  quite  unnecessary  ;  an  experienced  workman 
will  readily  form  the  curves  guided  alone  by  his  practised 
eye. 


FIG.  132. 

258. — Position  of  the  Balu§ters. — Place  the  centre  of 
the  first  baluster,  b  (Fig.  132),  half  its  diameter  from  the  face 
of  the  riser,  cd,  and  one  third  its  diameter  from  the  end  of 
the  step,  e  d\  and  place  the  centre  of  the  other  baluster,  a, 
half  the  tread  from  the  centre  of  the  first.  A  line  through 
the  centre  of  the  rail  will  occur  vertically  over  the  centres 
of  the  balusters.  The  usual  length  of  the  balusters  is  2  feet 
5  inches  and  2  feet  9  inches  respectively,  for  the  short  and 
long  balusters.  Their  length  may  be  greater  than  is  here 
indicated,  but,  for  safety,  should  never  be  less.  The  differ- 
ence in  length  between  the  short  and  long  balusters  is 
equal  to  one  half  the  height  of  a  riser. 


CONSTRUCTION   OF   WINDING  STAIRS. 


251 


259.— Winding  stairs:  have  the  steps  narrower  atone 
end  than  at  the  other.  In  some  stairs  there  are  steps  of 
parallel  width  incorporated  with  the  tapering  steps  ;  in  this 
case  the  former  are  called  flyers,  and  the  latter  winder:. 

260.— Regular  Winding  Stairs — In  Fig.  133,  abed  rep- 
resents the  inner  surface  of  the  wall  enclosing  the  space 
allotted  to  the  stairs,  a  e  the  length  of  the  steps,  and  efgk 
the  cylinder,  or  face  of  the  front-string.  The  line  a  e  is  given 
as  the  face  of  the  first  riser,  and  the  point  /  for  the  limit  of 


FIG.  133 


the  last.  Make  e  i  equal  to  18  inches,  and  upon  o,  with  o  i 
for  radius,  describe  the  arc  i  j  \  obtain  the  number  of  risers 
and  of  treads  required  to  ascend  to  the  floor  at  j,  according 
to  Art.  252,  and  divide  the  arc  ij  into  the  same  number  of 
equal  parts  as  there  are  to  be  treads  :  through  the  points  of 
division,  1,2,  3,  etc.,  and  from  the  wall-string  to  the  front- 
string,  draw  lines  tending  to  the  centre,  o :  then  these  lines 
will  represent  the  face  of  each  riser,  and  determine  the  form 
and  width  of  the  steps.  Allow  the  necessary  projection  for 
the  nosing  beyond  a  e,  which  should  be  equal  to  the  thick- 


STAIRS. 

ness  of  the  step,  and  then  a  e  I  k  will  be  the  dimensions  for 
each  step.  Make  a  pitch-board  for  the  wall-string  having  a  k 
for  the  tread,  and  the  rise  as  previously  ascertained :  with 
this  lay  out  on  a  thicknessed  plank  the  several  risers  and 
treads,  as  at  Fig.  128,  gauging  from  the  upper  edge  of  the 
string  for  the  line  at  which  to  set  the  pitch-board. 

Upon  the  back  of  the  string,  with  a  ij-inch  dado  plane, 
make  a  succession  of  grooves  ij  inches  apart,  and  parallel 
with  the  lines  for  the  risers  on  the  face.  These  grooves 
must  be  cut  along  the  whole  length  of  the  plank,  and  deep 
enough  to  admit  of  the  plank's  bending  around  the  curve 
abed.  Then  construct  a  drum,  or  cylinder,  of  any  com- 
mon kind  of  stuff,  made  to  fit  a  curve  with  a  radius  the 
thickness  of  the  string  less  than  oa  ;  upon  this  the  string  must 
be  bent,  and  the  grooves  filled  with  strips  of  wood,  called 
keys,  which  must  be  very  nicely  fitted  and  glued  in.  After 
it  has  dried,  a  board  thin  enough  to  bend  around  on  the  out- 
side of  the  string  must  be  glued  on  from  one  end  to  the 
other,  and  nailed  with  clout-nails.  In  doing  this,  be  careful 
not  to  nail  into  any  place  opposite  to  where  a  riser  or  step  is 
to  enter  on  the  face. 

After  the  string  has  been  on  the  drum  a  sufficient  time 
for  the  glue  to  set,  take  it  off,  and  cut  the  mortices  for  the 
steps  and  risers  on  the  face  at  the  lines  previously  made ; 
which  may  be  done  by  boring  with  a  centre-bit  half  through 
the  string,  and  nicely  chiselling  to  the  line.  The  drum  need 
not  be  made  to  extend  over  the  whole  space  occupied  by  the 
stairs,  but  merely  so  far  as  requisite  to  receive  one  piece  of 
the  wall-string  at  a  time ;  for  it  is  evident  that  more  than 
One  will  be  required.  The  front-string  may  be  constructed 
in  the  same  manner  ;  taking  e  I  instead  of  a  k  for  the  tread  of 
the  pitch-board,  dadoing  it  with  a  smaller  dado  plane,  and 
bending  it  on  a  drum  of  the  proper  size. 

261. Winding  Stairs :  Shape  and  Position  of  Timbers. — 

The  dotted  lines  in  Fig.  133  show  the  position  of  the  timbers 
as  regards  the  plan ;  the  shape  of  each  is  obtained  as  follows: 
In  Fig.  134,  the  line  i  a  is  equal  to  a  riser,  less  the  thickness 
ot  the  floor,  and  the  lines  2  m,  3  n,  4  <?,  5  /,  and  6  q  are  each 


TIMBERS   FOR   WINDING   STAIRS.  253 

equal  to  one  riser.  The  line  a  2  is  equal  to  a  m  in  Fig.  133, 
the  line  m  3  to  m  n  in  that  figure,  etc.  In  drawing  this 
figure,  commence  at  a,  and  make  the  lines  a  i  and  a  2  of  the 
length  above  s*pecified,  and  draw  them  at  right  angles  to 
each  other ;  draw  2  m  at  right  angles  to  a  2,  and  m  3  at 
right  angles  to  ;;/  2,  and  make  2  m  and  m  3  of  the  lengths 
as  above  specified ;  and  so  proceed  to  the  end.  Then 
through  the  points  i,  2,  3,  4,  5,  and  6  trace  the  line  \b\  upon 
the  points  i,  2,  3,  4,  etc.,  with  the  size  of  the  timber  for 
radius,  describe  arcs  as  shown  in  the  figure,  and  by  these 
the  lower  line  may  be  traced  parallel  to  the  upper.  This 
will  give  the  proper  shape  for  the  timber,  a  b,  in  Fig.  133  ; 
and  that  of  the  others  may  be  found  in  a  similar  manner.  In 
ordinary  cases,  the  shape  of  one  face  of  the  timber  will  be 
sufficient,  for  a  good  workman  can  easily  hew  it  to  its 
proper  level  by  that ;  but  where  great  accuracy  is  desirable, 
a  pattern  for  the  other  side  may  be  found  in  the  same  man- 


FIG.  134. 

ner  as  for  the  first.  In  many  cases,  the  timbers  beneath  cir- 
cular stairs  are  put  up  after  the  stairs  are  erected,  and  with- 
out previously  giving  them  the  required  form  ;  the  work- 
man in  shaping  them  being  guided  by  the  form  marked  out 
by  the  lower  edge  of  the  risers. 

262. — Winding  Stairs  with  Flyers  :  Grade  of  Front- 
String. — In  stairs  of  this  kind,  if  the  winders  are  confined  to 
the  quarter  circle,  the  transition  from  the  winders  to  the 
flyers  is  too  abrupt  for  convenience,  as  well  as  in  appear- 
ance. To  remove  this  unsightly  bend  in  the  rail  and  string, 
it  is  usual  to  take  in  among  the  winders  one  or  more  of  the 
flyers,  and  thus  graduate  the  width  of  the  winders  to  that  of 
the  flyers.  But  this  is  not  always  done  so  as  to  secure  the 
best  results.  By  the  method  now  to  be  shown,  both  rail  and 
strings  will  be  gracefully  graded.  In  Fig.  135,  a  b  repre- 
sents the  line  of  the  facia  along  the  floor  of  the  upper  story, 


254  STAIRS. 

bee  the  face  of  the  cylinder,  and  c  d  the  face  of  the  front- 
string.  Make  gb  equal  to  £  of  the  diameter  of  the  baluster, 
and  parallel  to  a  b,  b  e  c,  and  c  d  draw  the  centre-line  of  the 
rail,  fg>  g  k  z,  and  ij\  make  gk  and  gl  each  equal  to  half  the 
width  of  the  rail,  and  through  k  and  /,  parallel  to  the  centre- 
line, draw  lines  for  the  convex  and  the  concave  sides  of  the 
rail ;  tangical  to  the  convex  side  of  the  rail,  and  parallel  to 
k  m,  draw  ;/  o ;  obtain  the  stretch-out,  q  r,  of  the  semicircle, 
k p  m,  according  to  Art.  524;  extend  a  b  to  /,  and  k  m  to  s; 
make  c  s  equal  to  the  length  of  the  steps,  and  i  u  equal  to  18 
inches,  and  parallel  to  m  p  describe  the  arcs  s  t  and  u  6 ; 
from  /  draw  /  w,  tending  to  the  centre  of  the  cylinder ;  from 
6,  and  on  the  line  6  u  x,  run  off  the  regular  tread,  as  at  5,  4, 
3,  2,  i,  and  v\  make '  u  x  equal  to  half  the  arc  u  6,  and  make 
the  point  of  division  nearest  to  x,  as  v,  the  limit  of  the  par- 
allel steps,  or  flyers ;  make  r  o  equal  to  m  z;  from  o  draw  o 
a**  at  right  angles  to  ;/  o,  and  equal  to  one  riser;  from  a2 
draw  a*  s  parallel  to  n  o,  and  equal  to  one  tread;  from  s, 
through  o,  draw  s  b*. 

Then  from  w  draw  w  c2  at  right  angles  to ;/  o,  and  set  up 
on  the  line  w  c^  the  same  number  of  risers  that  the  floor,  A, 
is  above  the  first  winder,  B,  as  at  i,  2,  3,  4,  5,  and  6  ;  through 
5  (on  the  arc  6  u)  draw  d*  e*,  tending  to  the  centre  of  the 
cylinder;  from  e*~  draw  erf*  at  right  angles  to  110,  and 
through  5  (on  the  line  w  c*}  draw  g*  f2  parallel  to  n  o\ 
through  6  (on  the  line  zu  c2)  and/2  draw  the  line  h*  b*\ 
make  6  c~  equal  to  half  a  riser,  and  from  c*  and  6  draw  c'2  i'2 
and  6/2  parallel  to  n  o\  make  h*  i*  equal  to  7/2/2;  from  i- 
draw  i*  k*  at  right  angles  to  i*  h-,  and  from  f~  draw  /2  k* 
at  right  angles  to  /2  /*2;  upon  k~,  with  £2/2  for  radius,  de- 
scribe the  arc  /2  *2;  make  b~  /2  equal  to  £2/2,  and  ease  oft 
the  angle  at  b*  by  the  curve/2  /2.  In  the  figure,  the  curve 
is  described  from  a  centre,  but  as  this  might  be  imprac- 
ticable in  a  full-size  plan,  the  curve  may  be  obtained  accord- 


*  In  the  references  a2,  />',  etc.,  a  new  form  is  introduced  for  the  first  time. 
During  the  time  taken  to  refer  to  the  figure,  the  memory  of  the  form  of  these 
may  pass  from  the  mind,  while  that  of  the  sound  alone  remains;  they  may 
then  be  mistaken  for  a  2,  b  2,  etc.  This  can  be  avoided  in  reading  by  giving 
them  a  sound  corresponding  to  their  meaning,  which  is  a  second,  b  second,  etc. 


MOULDS   FOR   QUARTER-CIRCLE   STAIRS. 


255 


ing  to  Art.  521.  Then  from  i,  2,  3,  and  4  (on  the  line  w  c~) 
draw  lines  parallel  to  n  o,  meeting  the  curve  in  w2,  n~,  o2, 
and  /2;  from  these  points  draw  lines  at  right  angles  to  ;/  o, 
and  meeting  it  in  jr2,  r2,  sz,  and  /2;  from  x~  and  r2  draw 


FIG.  135. 

lines  tending  to  z/2,  and  meeting  the  convex  side  of  the  rail 
in  j/2  and  £2 ;  make  m  ?>2  equal  to  r  s~,  and  m  w*  equal  to 
rt*-,  fromj/ W2,  and  w2,  through  4,  3,  2,  and  i,  draw  lines 
meeting  the  line  of  the  wall-string  in  a3,  £3,  c\  and  rt78 ;  from 


256  STAIRS. 

e3,  where  the  centre-line  of  the  rail  crosses  the  line  of  the 
floor,  draw  e3/3  at  right  angles  to  n  o,  and  from/3,  through 
6,  draw  f*g~\  then  the  heavy  lines  f*g\  e*  d\  y*  a*,  Z*  b\ 
v*c3,  w2  d3,  and  z y  will  be  the  lines  for  the  risers,  which, 
being  extended  to  the  line  of  the  front-string,  b  e  c  d,  will 
give  the  dimensions  of  the  winders  and  the  grading  of  the 
front-string,  as  was  required. 

HAND-RAILING. 

263.— Hand-Railing  for  Stairs.— A  piece  of  hand-rail- 
'  ing  intended  for  the  curved  part  of  a  stairs,  when  properly 
shaped,  has  a  twisted  form,  deviating  widely  from  plane  sur- 
faces. If  laid  upon  a  table  it  may  easily  be  rocked  to  and 
fro,  and  can  be  made  to  coincide  with  the  surface  of  the 
table  in  only  three  points.  And  yet  it  is  usual  to  cut  such 
twisted  pieces  from  ordinary  parallel-faced  plank ;  and  to 
cut  the  plank  in  form  according  to  a  face-mould,  previously 
formed  from  given  dimensions  obtained  from  the  plan  of  the 
stairs.  The  shape  of  the  finished  wreath  differs  so  widely 
from  the  piece  when  first  cut  from  the  plank  as  to  make  it 
appear  to  a  novice  a  matter  of  exceeding  difficulty,  if  not  an 
impossibility,  to  design  a  face-mould  which  shall  cover  accu- 
rately the  form  of  the  completed  wreath.  But  he  will  find, 
as  he  progresses  in  a  study  of  the  subject,  that  it  is  not  only 
a  possibility,  but  that  the  science  has  been  reduced  to  such 
a  system  that  all  necessary  moulds  may  be  obtained  with 
great  facility.  To  attain  to  this  proficiency,  however,  re- 
quires close  attention  and  continued  persistent  study,  yet  no 
more  than  this  important  science  deserves.  The  young  car- 
penter may  entertain  a  less  worthy  ambition  than  that  of 
desiring  to  be  able  to  form  from  planks  of  black-walnut  or 
mahogany  those  pieces  of  hand-railing  which,  when  secured 
together  with  rail-screws,  shall,  on  applying  them  over  the 
stairs  for  which  they  are  intended,  be  found  to  fit  their 
places  exactly,  and  to  form  graceful  curves  at  the  cylinders. 
That  railing  which  requires  to  be  placed  upon  the  stairs 
before  cutting  the  joints,  or  which  requires  the  curves  or 
butt-joints  to  be  refitted  after  leaving  the  shop,  is  discredit- 


PRINCIPLES   OF   HAND-RAILING.  257 

able  to  the  workman  who  makes  it.  No  true  mechanic  will 
be  content  until  he  shall  be  proved  able  to  form  the  curves 
and  cut  the  joints  in  the  shop,  and  so  accurately  that  no  altera- 
tion shall  be  needed  when  the  railing  is  brought  to  its  place 
on  the  stairs.  The  science  of  hand-railing  requires  some 
knowledge  of  descriptive  geometry— that  branch  of  geometry 
which  has  for  its  object  the  solution  of  problems  involving 
three  dimensions  by  means  of  intersecting  planes.  The 
method  of  obtaining  the  lengths  and  bevils  of  hip  and  valley 
rafters,  etc.,  as  in  Art.  233,  is  a  practical  example  of  descrip- 
tive geometry.  The  lines  and  angles  to  be  developed  in 
problems  of  hand-railing  are  to  be  obtained  by  methods 
dependent  upon  like  principles. 

264.—  Hand-Railing:   Definitions;   Planes  and  Solids. 

—Preliminary  to  an  exposition  of  the  method  for  drawing 
the  face-moulds  of  a  hand-rail  wreath,  certain  terms  used  in 
descriptive  geometry  need  to  be  denned.  Among  the  tools 
used  by  a  carpenter  are  those  well-known  implements  called 
planes,  such  as  the  jack-plane,  fore-plane,  smoothing-plane, 
etc.  These  enable  the  workman  to  straighten  and  smooth 
the  faces  of  boards  and  plank,  and  to  dress  them  out  of 
wind,  or  so  that  their  surfaces  shall  be  true  and  unwinding. 
The  term  plane,  as  used  in  descriptive  geometry,  however, 
refers  not  to  the  implement  aforesaid,  but  to  the  unwinding 
surface  formed  by  these  implements.  A  plane  in  geometry 
is  defined  to  be  such  a  surface  that  if  any  two  points  in  it  be 
joined  by  a  straight  line,  this  line  will  be  in  contact  with  the 
surface  at  every  point  in  its  length.  With  like  results  lines 
may  be  drawn  in  all  possible  directions  upon  such  a  sur- 
face. This  can  be  done  only  upon  an  unwinding  surface ; 
therefore,  a  plane  is  an  unwinding  surface.  Planes  are 
understood  to  be  unlimited  in  their  extent,  and  to  pass  freely 
through  other  planes  encountered. 

The  science  of  stair-building  has  to  do  with  prisms  and 
cylinders,  examples  of  which  are  shown  in  Figs.  136,  137,  and 
138.  A  right  prism  (Figs.  136  and  137)  is  a  solid  standing 
upon  a  horizontal  plane,  and  with  faces  each  of  which  is  a 
plane.  Two  of  these  faces— top  and  bottom— are  horizontal 


258 


STAIRS. 


and  are  equal  polygons,  having  their  corresponding  sides 
parallel. 

The  other  faces  of  the  prism  are  parallelograms,  each  of 
which -is  a  vertical  plane.  When  the  vertical  sides  of  a 
prism  are  of  equal  width,  and  in  number  increased  indefi- 
nitely, the  two  polygonal  faces  of  the  prism  do  not  differ 
essentially  from  circles,  and  thence  the  prism  becomes  a 
cylinder.  Thus  a  right  cylinder  may  be  defined  to  be  a 
prism,  with  circles  for  the  horizontal  faces  (Fig.  138). 


FIG.  136. 


FIG.  137- 


FIG.  138. 


265.  —  Hand  -  Railing  :    Preliminary  Considerations.  — 

If  within  the  well-hole,  or  stair-opening,  of  a  circular  stairs  a 
solid  cylinder  be  constructed  of  such  diameter  as  shall  fill  the 
well-hole  completely,  touching  the  hand-railing  at  all  points, 
and  then  if  the  top  of  this  cylinder  be  cut  off  on  a  line  with 
the  top  of  the  hand-railing,  the  upper  end  of  the  cylinder 
would  present  a  winding  surface.  But  if,  instead  of  cutting 
the  cylinder  as  suggested,  it  be  cut  by  several  planes,  each 
of  which  shall  extend  so  as  to  cover  only  one  of  the  wreaths 
of  the  railing,  and  be  so  inclined  as  to  touch  its  top  in  three 
points,  then  the  form  of  each  of  these  planes,  at  its  intersec- 
tion with  the  vertical  sides  of  the  cylinder,  would  present 
the  shape  of  the  concave  edge  of  the  face-mould  for  that 
particular  piece  of  hand  -  railing  covered  by  the  plane. 
Again,  if  a  hollow  cylinder  be  constructed  so  as  to  be  in 
contact  with  the  outer  edge  of  the  hand-railing  throughout 
its  length,  and  this  cylinder  be  also  cut  by  the  aforesaid 


FACE-MOULDS   FOR   HAND-RAILS.  259 

planes,  then  each  of  said  planes  at  its  intersection  with  this 
latter  cylinder  would  present  the  form  of  the  convex  edge 
of  the  said  face-mould.  A  plank  of  proper  thickness  may 
now  have  marked  upon  it  the  shape  of  this  face-mould,  and 
the  piece  covered  by  the  face-mould,  when  cut  from  the 
plank,  will  evidently  contain  a  wreath  like  that  over  which 
the  face-mould  was  formed,  and  which,  by  cutting  away  the 
surplus  material  above  and  below,  may  be  gradually  wrought 
into  the  graceful  form  of  the  required  wreath. 

By  the  considerations  here  presented  some  general  idea 
may  be  had  of  the  method  pursued,  by  which  the  form  of  a 
face-mould  for  hand-railing  is  obtained.  A  little  reflection 
upon  what  has  been  advanced  will  show  that  the  problem 
to  be  solved  is  to  pass  a  plane  obliquely  through  a  cylinder 
at  certain  given  points,  and  find  its  shape  at  its  intersection 
with  the  vertical  surface  of  \he  cylinder.  Peter  Nicholson 
was  the  first  to  show  how  this  might  be  done,  and  for  the 
invention  was  rewarded,  by  a  scientific  society  of  London, 
with  a  gold  medal.  Other  writers  have  suggested  some 
slight  improvements  on  Nicholson's  methods.  The  method 
to  which  preference  is  now  given,  for  its  simplicity  ot  work- 
ing and  certainty  of  results,  is  that  which  deals  with  the 
tangents  to  the  curves,  instead  of  with  the  curves  themselves; 
so  we  do  not  pass  a  plane  through  a  cylinder,  but  through  a 
prism  the  vertical  sides  of  which  are  tangent  to  the  cylinder, 
and  contain  the  controlling  tangents  of  the  face-moulds.  The 
task,  therefore,  is  confined  principally  to  finding  the  tangents 
upon  the  face-mould.  This  accomplished,  the  rest  is  easy,  as 
will  be  seen. 

The  method  by  which  is  found  the  form  of  the  top  of  a 
prism  cut  by  an  oblique  plane  will  now  be  shown. 

266.— A  Pri§m  Cut  toy  an  Oblique  Plane — A  prism  is 
shown  in  perspective  at  Fig.  139,  cut  by  an  oblique  plane. 
The  points  abed  are  the  angles  of  the  horizontal  base,  and 
abg,  bcf,  cdcf,  and  adeg  are  the  vertical  sides;  while 
efbg  is  the  top,  the  form  of  which  is  to  be  shown. 

267.— Form  of  Top  of  Pri§m — In  Fig.  139  the  form  of 
the  top  of  the  prism  is  shown  as  it  appears  in  perspective.. 


26o 


STAIRS. 


not  in  its  real  shape  ;  this  is  now  to  be  developed.     In  Fig. 
140,  let  the  sauare  a  b  c  d  represent  by  scale  the  actual  form 


FIG.  139. 


and  size  of  the  base,  a  b  cd,  of  the  prism  shown  in  Fig.  1  39. 
Make  c  c,  and  ddt  respectively  equal  to  the  actual  heights  at 


FIG.  140. 

cf  and  de,  Fig.  139  ;  the  lines  ddt  and  c  c,  being  set  up  per- 
pendicular to  the  line  dc.    Extend  the  lines  dc  and  dtct  until 


ILLUSTRATION   BY   PLANES.  26 1 

they  meet  in  h  ;  join  b  and  h.  Now  this  line  b  h  is  the  inter- 
section of  two  planes :  one,  the  base,  or  horizontal  plane  upon 
which  the  prism  stands ;  the  other,  the  cutting  plane,  or  the 
plane  which,  passing-  obliquely  through  the  prism,  cuts  it  so 
as  to  produce,  by  intersecting  the  vertical  sides  of  the  prism, 
the  form  b  fe g,  Fig.  139. 

To  show  that  b  k  is  the  line  of  intersection  of  these  two 
planes,  let  the  paper  on  which  the  triangle  dhdt  is  drawn 
(designated  by  the  letter  B)  be  lifted  by  the  point  dt  and 
revolved  on  the  line  dk  until  dt  stands  vertically  over  d,  and 
ct  over  c\  then  B  will  be  a  plane  standing  on  the  line  dh, 
vertical  to  the  base-plane  A.  The  point  h  being  in  the  line 
cd  extended,  and  the  line  cd  being  in  the  base-plane  A,  there- 
fore h  is  in  the  base-plane  A.  Now  the  line  dtct  represents 
the  line  cf  of  Fig.  139,  and  is  therefore  in  the  cutting  plane  ; 
consequently  the  point  //,  being  also  in  the  line  dt  c,  ex- 
tended, is  also  in  the  cutting  plane.  By  reference  to  Fig. 
139  it  will  be  seen  that  the  point  b  is  in  both  the  cutting  and 
base  planes ;  we  must  therefore  conclude  that,  since  the  two 
points  b  and  h  are  in  both  the  cutting  and  base  planes,  a  line 
joining  these  two  points  must  be  the  intersection  of  these  two 
planes.  The  determination  of  the  line  of  intersection  of  the 
base  and  cutting  planes  is  very  important,  as  it  is  a  control- 
ling line  ;  as  will  be  seen  in  denning  the  lines  upon  which 
the  form  of  the  face-mould  depends.  Care  should  therefore 
be  taken  that  the  method  of  obtaining  it  be  clearly  under- 
stood. 

It  will  be  observed  that  the  intersecting  line  bh,  being  in 
the  horizontal  plane  A,  is  therefore  a  horizontal  line.  Also, 
that  this  horizontal  line  b  h  being  a  line  in  the  cutting  plane, 
therefore  all  lines  upon  the  cutting  plane  which  are  drawn 
parallel  to  b  h  must  also  be  horizontal  lines.  The  import- 
ance of  this  will  shortly  be  seen.  Through  a,  perpendicular 
to  bh,  draw  the  line  bnd^  and  parallel  with  this  line  draw 
ddini ;  on  d  as  centre  describe  the  arc  dt  dltil ;  draw  dltll  dv 
parallel  with  ddtlJ  and  extend  the  latter  to  dtll ;  on  d{l  as 
centre  describe  the  arc  dv  dnl ;  join  bn  and  dul.  We  now 
have  three  vertical  planes  which  are  to  be  brought  into 
position  around  the  base-plane  A,  'as  follows:  Revolve  B 


262  STAIRS. 

upon  dh,  E  upon  ddit,  and  C  upon  bn  dtn  each  until  it  stands 
perpendicular  to  the  plane  -A.  Then  the  points  dt  and  dillt 
will  coincide  and  be  vertically  over  d\  the  points  dllt  and  dv 
will  coincide  and  stand  vertically  over  dn  ;  and  ct  will  cover  c. 
These  vertical  planes  will  enclose  a  wedge-shaped  figure, 
lying  with  one  face,  b^d^dh,  horizontal  and  coincident  with 
the  base-plane  A,  and  three  vertical  faces,  blt  du  dti/y  ddn  d^  djiit, 
and  hddt.  By  drawing  the  figure  upon  a  piece  of  stout 
paper,  cutting  it  out  at  the  outer  edges,  making  creases  in 
the  lines  hd,  ddtl,  dubt/J  then  folding  the  three  planes  B,  E, 
and  C  at  right  angles  to  A,  the  relation  of  the  lines  will  be 
readily  seen.  Now,  to  obtain  the  form  of  the  top  or  cover  to 
the  wedge-shaped  figure,  perpendicular  to  bn  diti  draw  b,,h, 
and  dtlle\  on  btl  as  centre  describe  the  arc  hhi  ;  make  dllte 
equal  to  dlt  d\  join  e  and  ht.  Now  the  form  of  the  top  of  the 
wedge-shaped  figure  is  shown  within  the  bounds  din  bi{  ht  c. 
By  revolving  this  plane  D  on  the  line  bn  dllt  until  it  is  at 
a  right  angle  to  the  plane  C,  and  this  while  the  latter  is 
supposed  to  be  vertical  to  the  plane  A,  it  will  be  perceived 
that  this  movement  will  place  the  plane  D  on  top  of  the 
wedge-shaped  figure,  and  in  such  a  manner  as  that  the  point 
e  will  coincide  with  dlllt  d{,  and  the  point  ht  will  fall  upon  and 
be  coincident  with  the  point  h,  and  the  lines  of  the  cover 
will  coincide  with  the  corresponding  lines  of  the  top  edges 
of  the  sides  of  the  figure;  for  example,  the  line  blidlll  is 
common  to  the  top  and  the  side  C;  the  line  dltle  equals  dtl  d, 
which  equals  dvdttil\  therefore,  the  line  ditte  will  coincide 
with  dvdi/u  of  the  side  E\  the  line  eht  will  coincide  with  d,h 
of  the  side  B\  and  the  line  bl,hl  will  coincide  with  the  line 
btlh.  Thus  the  figure  D  bounded  by  blldlllchl  will  exactly 
fit  as  a  cover  to  the  wedge-shaped  figure.  Upon  this  cover 
we  may  now  develop  the  form  of  the  top  of  the  prism. 

Preliminary  thereto,  however,  it  will  be  observed,  as  was 
before  remarked,  that  lines  upon  the  cutting  plane  which 
are  parallel  to  the  intersecting  line  btl  ht  are  horizontal ; 
and  each,  therefore,  must  be  of  the  same  length  as  the  line 
in  the  base-plane  A  vertically  beneath  it.  For  example,  the 
line  dllt  e{  is  a  line  in  the  cutting  plane  D,  parallel  with  the 
line  bltht  in  the  same  plane,  and  this  line  blllil  will  (when  the 


EXPLANATION   OF   THE  DIAGRAMS.  263 

cutting  plane  D  is  revolved  into  its  proper  position)  be  co- 
incident with  the  intersecting  line  blt  h  ;  therefore,  the  line 
dtile  is  a  line  in  the  cutting  plane  D,  drawn  parallel  with  the 
intersecting  line  bu  h.  Now  this  line  dllt  e,  when  in  position, 
will  be  coincident  with  the  line  dltlld^  which  lies  vertically 
over  the  line  dt,d-ol  the  base-plane  A  ;  its  length,  therefore, 
is  equal  to  that  of  the  latter.  In  like  manner  it  may  be 
shown  that  the  length  of  any  line  on  the  plane  D  parallel 
to  btl  hn  is  equal  in  length  to  the  corresponding  line  upon 
the  plane  A  vertically  beneath  it. 

Therefore,  to  obtain  the  form  of  the  top  of  the  prism,  we 
proceed  as  follows :  Perpendicular  to  btl  dv  draw  c  clu  and 
aattl\  perpendicular  to  btldni  draw  ctllf  and  equal  toc,,c; 
on  blt  as  centre  describe  the  arc  b  bt ;  join  bt  a,n,  btf,  and  al4l  e. 
Now  we  have  here  in  plane  D  the  form  of  the  top  of  the 
prism,  as  shown  in  the  figure  bounded  by  the  lines  a^'fije. 
This  will  be  readily  seen  when  the  plane  D  is  revolved  into 
position.  Then  the  point  atll  will  be  vertically  over  a  ;  the 
point  e  coincident  with  dt  dltll  and  vertically  over  d;  the  point 
/  coincident  with  c/  and  vertically  over  c ;  while  bt  will  coin- 
cide with  b  of  the  base-plane  A. 

The  figure  ani  btfe,  therefore,  represents  correctly  both 
in  form  and  size  the  top  of  the  prism  as  it  is  shown  in  per- 
spective at  bfeg,  Fig.  139.  The  line  ef,  Fig.  140,  is  equal  to 
the  line  dt  ct,  and  so  of  the  other  lines  bounding  the  edges  of 
the  figure. 

The  cutting  plane  b  f  e  g,  Fig.  139,  may  be  taken  to  repre- 
sent the  surface  of  the  plank  from  which  the  wreath  of  hand- 
railing  is  to  be  cut ;  the  wreath  curving  around  from  b  to  ct 
as  shown  in  Fig.  141,  the  lines  b  g  and  ge  being  tangent  to 
the  curve  in  the  cutting  plane;  while  ab  and  ad  are  tan- 
gents to  the  curve  on  the  base  plane,  or  plane  of  the  cylin- 
der. The  location  of  the  cutting  plane,  however,  is  usually 
not  at  the  upper  surface  of  the  plank,  but  midway  between 
the  upper  and  under  surfaces.  The  tangents  in  the  plane 
are  found  to  be  more  conveniently  located  here  for  deter- 
mining the  position  of  the  butt-joints.  For  a  moulded  rail 
two  curved  lines,  each  with  a  pair  of  tangents,  are  required 
upon  the  cutting  plane,  one  for  the  outer  edge  of  the  rail, 


264  STAIRS. 

and  the  other  for  the  inner  edge  ;  but  for  a  round  rail  only 
one  curve  with  its  tangents  is  required,  as  that  from  b  to  e 
in  Fig.  141,  which  is  taken  to  represent  the  curved  line  run- 
ning through  the  centre  of  the  cross-section  of  the  rail.  As 
an  easy  application  of  the  principles  regarding  the  prism, 
just  developed,  an  example  will  now  be  given. 


268. — Face-Mould  for  Hand-Railing  of  Platform  Stairs. 

— Let/£  and  /  ;;/,  Fig.  142,  represent  the  central  or  axial  lines 
of  the  hand-rails  of  the  two  flights,  one  above,  the  other  be- 
low the  platform ;  and  let  the  semicircle/df/  be  the  central 
line  of  the  rail  around  the  cylinder  at  the  platform,  the  risers 
at  the  platform  being  located  at  j  and  /.  Vertically  over  the 
platform  risers  draw  ggt ;  make  grt  equal  to  a  riser  of  the 
lower  flight,  and  rtgt  and  sst  each  equal  to  a  riser  of  the 
upper  flight.  Draw  gt  s  and  gkt  horizontal  and  equal 
each  to  a  tread  of  each  flight  respectively.  Through  r,  draw 
k,  au,  and  through  gy  draw  st  tt.  Vertically  over  d  draw  at  tr 
Horizontally  draw  atl  anll  and  tt  ttl. 

It  is  usual  to  extend  the  wreath  of  the  cylinder  so  as  to 
include  a  part  of  the  straight  rail — such  a  part  as  convenience 
may  require.  Let  the  straight  part  here  to  be  included  ex- 
tend from  /  to  b  on  the  plan.  Vertically  over  b  draw  bt  ctlli, 
and  horizontally  draw  b/  w-tl ;  at  any  point  on  bt  wtt  locate  wn, 
and  make  wtl  wf  equal  to  j  I,  and  bisect  it  in  w;  erect  the 
perpendiculars  ivt  altil,  w  dv/J,  and  w/7  v  ;  join  tlt  and  atlll ;  from 
dvil  horizontally  draw  dvil  dv/ ;  parallel  with  rt  kt  draw 
dv,  ctlll'  We  now  have  the  plan  and  elevations  of  the  prism, 


FACE-MOULD  FOR   PLATFORM   STAIRS. 


265 


containing  at  its  angles  the  tangents  required  for  the  wreath 
extending  from  b  to  d  on  the  plan.  The  elevation  F  is  a  view 
of  the  cylinder  looking  in  the  direction  dc. 


FIG.  142. 


Comparing  Fig.   142  with  Fig.  141*  the  line  b,  w/t  is  the 
trace,  upon  a  vertical  plane,  of  the  horizontal  plane  abed 


266  STAIRS. 

of  Fig.  141,  or  is  the  ground-line  from  which  the  heights  of 
the  prism  are  to  be  taken. 

The  triangle^  bt  au  is  represented  in  Fig.  141  at  ab g,  and 
the  inclined  line  b,  atl  is  the  tangent  of  the  rail  of  the  lower 
flight,  and  is  represented  in  Fig.  141  at  bg ;  while  aniltn  is 
the  tangent  of  the  railing  around  the  cylinder,  and  the  half 
of  it  is  represented  in  Fig.  141  at  ge.  The  height  btctilJ  is 
shown  in  Fig.  141  at  c  f,  while  the  height  iv  dv,t,  or  a/dVl,  is 
shown  in  Fig.  141  at  de. 

The  vertical  planes  EEC  may  now  be  constructed  about 
the  prism  as  in  Fig.  140,  proceeding  thus :.  Make  c  cf  equal  to 
bt  ciul,  and  ddt  equal  to  at  dv/ ;  through  ct  draw  di  h\  through 
b  draw  h  btl ;  perpendicular  to  h  blt  through  a  draw  bn  dv ; 
from  ^parallel  with  bn  */vdraw  d  djiti  ;  on  d  as  centre  describe 
the  arc  dtdnil\  draw  ditll  dv,  also  d  dltl,  parallel  with  hbn\ 
on  dti  as  centre  describe  the  arc  dvdtj  ;  join  dw  to  bjt.  Par- 
allel with  bn  h  draw  from  each  important  point  of  the  plan, 
as  shown,  an  ordinate  extending  to  the  line  btl  dtji,  and  thence 
across  plane  D  draw  ordinates  perpendicular  to  blt  dtll,  and 
make  them  respectively  equal  to  the  corresponding  ordinates 
of  the  plane  A,  measured  from  the  line  bn  dv;  join  e  to/,  all{ 
to  b^  altl  to  e,  and  b,  to/;  also  join  /,  to  rt.  Then  ain  b,  is  the 
tangent  standing  over  a  b,  and  alit  e  is  Jthe  tangent  standing 
over  ad.  The  line  blli  is  the  part  of  the  tangent  which 
stands  over  blt,  the  portion  of  the  wreath  which  is  straight. 
The  curve  enlplll  is  the  trace  upon  the  cutting  plane  of  the 
quarter  circle  dnpl,  traced  through  the  points  /*,/,,  and  as 
many  more  as  desirable,  found  by  ordinates  as  any  other 
point  in  the  plane  A.  Thus  we  have  complete  the  line 
bt  I,  nt  e,  the  central  line  of  the  wreath  extending  from  b  to  d 
in  the  plan.  This  is  the  essential  part  of  the  face-mould,  which 
is  now  to  be  drawn  as  follows:  At  Fig.  143  repeat  the  par- 
allelogram atll  btfe  of  Fig.  142,  and,  with  a  radius  equal  to 
half  the  diameter  of  the  rail,  describe,  from  centres  taken  on 
the  central  line,  the  several  circles  shown ;  and  tangent  to 
these  circles  draw  the  outer  and  inner  edges  of  the  rail. 
The  joint  at  bt  is  to  be  drawn  perpendicular  to  the  tangent 
b.  ain,  while  that  at  e  is  to  be  perpendicular  to  the  tangent 
^atll.  This  completes  the  face-mould  for  the  wreath  over 


WREATHS   FOR   A   ROUND   RAIL.  267 

bind  of  the  plan.  If  the  pitch-board  of  the  upper  flight  be 
the  same  as  that  of  the  lower  flight,  the  face-mould  at  Fig. 
143  will,  reversed,  serve  also  for  the  wreath  over  the  other 
half  of  the  cylinder. 

In  using  this  face-mould,  place  it  upon  a  plank  equal  in 
thickness  to  the  diameter  of  the  rail,  mark  its  form  upon  the 
plank,  and  saw  square  through ;  then  chamfer  the  wreath  to 
an  octagonal  form,  after  which  carefully  remove  the  angles 
so  as  to  produce  the  required  round  form.  The  joints,  as  well 
as  the  curved  edges,  are  to  be  cut  square  through  the  plank. 

Many  more  lines  have  been  used  in  obtaining  this  face- 
mould  than  were  really  necessary  for  so  simple  a  case,  but  no 
more  than  was  deemed  advisable  in  order  properly  to  eluci- 
date the  general  principles  involved.  A  very  simple  method 


FIG.  143. 

for  face-moulds  of  platform  stairs  with  small  cylinders  will 
now  be  shown. 

269. — More  Simple  method  for  Hand-Rail  to  Platform 
Stairs. — In  Fig.  144,  jge  represents  a  pitch-board  of  the  first 
flight,  and  d  and  i  the  pitch-board  of  the  second  flight  of  a  plat- 
form stairs,  the  line  e  f  being  the  top  of  the  platform  ;  and 
abc  is  the  plan  of  a  line  passing  through  the  centre  of 
the  rail  around  the  cylinder.  Through  i  and  d  draw  i  k, 
and  through  y  and  e  drawy  k  ;  from  k  draw  k  I  parallel  to/e; 
from  b  draw  bm  parallel  to  gd ;  from  /  draw  Ir  parallel  to 
kj ;  from  n  draw  nt  at  right  Angles  toy/6;  on  the  line  ob 
make  ot  equal  to  nt ;  join  c  and  t ;  on  the  line  jc,  Fig.  145, 
make  ec  equal  to  en  at  Fig.  144;  from  c  draw  c  t  at  right 
angles  to  j  c,  and  make  ct  equal  to  c  t  at  Fig.  144;  through  / 
draw  p  I  parallel  to  j  c,  and  make  //  equal  to  /  /  at  Fig.  144 ; 
join  /and  c,  and  complete  the  parallelogram  eels  ;  find  the 
points  o,  o,  o,  according  to  Art.  551  ;  upon  e,  o,  o,  o,  and  /, 


268 


STAIRS. 


successively,  with  a  radius  equal  to  half  the  width  of  the 
rail,  describe  the  circles  shown  in  the  figure ;  then  a  curve 
traced  on  both  sides  of  these  circles,  and  just  touching  them, 


FIG,  144. 

Avill  give  the  proper  form  for  the  mould, 
drawn  at  right  angles  to  c  /. 


The  joint  at  /  is 


FIG.  145. 

This  simple  method  for  obtaining  the  face-moulds  for  the 
hand-rail  of  a  platform  stairs  appeared  first  in  the  early  edi- 
tions of  this  work.  It  was  invented  by  a  Mr.  Kells,  an 


HAND-RAIL  TO    PLATFORM   STAIRS. 


269 


eminent  stair-builder  of  this  city.  A  comparison  with  Fig. 
142  will  explain  the  use  of  the  few  lines  introduced.  For  a 
full  comprehension  of  it  reference  is  made  to  Fig.  146,  in 
which  the  cylinder,  for  this  purpose,  is  made  rectangular 


FIG.  146. 

instead  of  circular.  ,The  figure  gives  a  perspective  view  of 
a  part  of  the  upper  and  of  the  lower  flights,  and  a  part  of 
the  platform  about  the  cylinder.  The  heavy  lines,  ////,  me, 
and  cj,  show  the  direction  of  the  rail,  and  are  supposed  to 
pass  through  the  centre  of  it.  Assuming  that  the  rake  of 


270 


STAIRS. 


the  second  flight  is  the  same  as  that  of  the  first,  as  is  gener- 
ally the  case,  the-  face-mould  for  the  lower  twist  will,  when 
reversed,  do  for  the  upper  flight ;  that  part  of  the  rail,  there- 
fore, which  passes  from  e  to  c,  and  from  c  to  /,  is  all  that  will 
need  explanation. 

Suppose,  then,  that  the  parallelogram  eaoc  represent  a 
plane  lying  perpendicularly  over  'eabf,  being  inclined  in 
the  direction  ec,  and  level  in  the  direction  co  ;  suppose  this 


FIG.  147. 

plane  eaoc  be  revolved  on  ec  as  an  axis,  in  the  manner  indi- 
cated by  the  arcs  o  n  and  a  x,  until  it  coincides  with  the 
plane  ertc\  the  line  ao  will  then  be  represented  by  the  line 
x  n  ;  then  add  the  parallelogram  xrtn,  and  the  triangle  ctl, 
deducting  the  triangle  ers\  then  the  edges  of  the  plane  cslc, 
inclined  in  the  direction  ec,  and  also  in  the  direction  c  I,  will 
lie  perpendicularly  over  the  plane  eabf.  From  this  we 
gather  that  the  line  co,  being  at  right  angles  to  ec,  must, -in 


HAND-RAIL   FOR   LARGE   CYLINDER.  2JI 

order  to  reach  the  point  /,  be  lengthened  the  distance  nt, 
and  the  right  angle  ect  be  made  obtuse  by  the  addition  to 
it  of  the  angle  tc  /.  By  reference  to  Fig.  144,  it  will  be  seen 
that  this  lengthening  is  performed  by  forming  the  right- 
angled  triangle  cot,  corresponding  to  the  triangle  cot  in 
Fig.  146.  The  line  ct  is  then  transferred  to  Fig.  145,  and 
placed  at  right  angles  tov^r;  this  angle  ect  is  then  increased 
by  adding  the  angle  tcl,  corresponding  to  tcl,  Fig.  146. 
Thus  the  point  /  is  reached,  and  the  proper  position  and 
length  of  the  lines  ec  and  ^/obtained.  To  obtain  the  face- 
mould  for  a  rail  over  a  cylindrical  well-hole,  the  same  process 
is  necessary  to  be  followed  until  the  length  and  position  of 
these  lines  are  found ;  then,  by  forming  the  parallelogram 
eels,  and  describing  a  quarter  of  an  ellipse  therein,  the 
proper  form  will  be  given. 


FIG.  148. 

270. — Hand-Railling  for  a  Larger  Cylinder. — Fig.  147 
represents  a  plan  and  a  vertical  section  of  a  line  passing 
through  the  centre  of  the  rail  as  before.  From  b  draw  bk 
parallel  to  cd\  extend  the  lines  zWandyV  until  they  meet  kb 
in  k  and  /;  from  ;/  draw  nl  parallel  to  ob;  through  /  draw 
//  parallel  to  j  k;  from  k  draw  kt  at  right  angles  to/£;  on 
the  line  ob  make  ot  equal  to  kt.  Make  ec  (Fig.  148)  equal 
to  ek  at  Fig.  147  ;  from  c  draw  ct  at  right  angles  to  ec,  and 
equal  to  ct  at  Fig.  147  ;  from  /  draw  //  parallel  to  ce,  and 
make  tl  equal  to  //at  Fig.  147  ;  complete  the  parallelogram 
eels,  and  find  the  points  o,  0,  o,  as  before  ;  then  describe  the 
circles  and  complete  the  mould  as  in  Fig.  145.  The  difference 
between  this  and  Case  I  is  that  the  line  ct,  instead  of  being 
raised  and  thrown  out,  is  lowered  and  drawn  in.  A  method 
of  planning  a  cylinder  so  as  to  avoid  the  necessity  of  cant- 
ing the  plank,  either  up  or  down,  will  now  be  shown. 


2/2 


STAIRS. 


271. — Faee-moMld  without  Canting  the  Plank. —  Instead 
of  placing  the  platform-risers  at  the  spring  of  the  cylinder,  a 
more  easy  and  graceful  appearance  may  be  given  to  the 
rail,  and  the  necessity  of  canting  either  of  the  twists  entirely 
obviated,  by  fixing  the  place  of  the  above  risers  at  a  certain 
distance  within  the  cylinder,  as  shown  in  Fig.  149 — the  lines 
indicating  the  face  of  the  risers  cutting  the  cylinder  at  k  and 
/,  instead  of  at  /  and  q,  the  spring  of  the  cylinder.  To 
ascertain  the  position  of  the  risers,  let  abc  be  the  pitch- 
board  of  the  lower  flight,  and  cde  that  of  the  upper  flight, 

these  being  placed  so  that  b  c  and 
cd  shall  form  a  right  line.  Extend 
a  c  to  cut  de  in  f;  draw  fg  parallel 
*to  db,  and  of  indefinite  length ; 
draw  go  at  right  angles  to  fg,  and 
equal  in  length  to  the  radius  of  the 
circle  formed  by  the  centre  of  the 
rail  in  passing  around  the  cylinder  ; 
on  o  as  centre  describe  the  semi- 
circle/^z'/  through  o  draw  is  par- 
allel to  db;  make  oh  equal  to  the 
radius  of  the  cylinder,  and  describe 
on  o  the  face  of  the  cylinder  phq; 
then  extend  db  across  the  cylinder, 
cutting  it  in  /  and  k — giving  the 
position  of  the  face  of  the  risers, 
as  required.  To  find  the  face- 
mould  for  the  twists  is  simple  and 
obvious :  it  being  merely  a  quarter 
of  an  ellipse,  having  oj  for  semi- 
minor  axis,  and  sf  for  the  semi-major  axis;  or,  at  Fig.  151, 
let  dci"be<\.  right  angle  ;  make  c  i  equal  to  oj,  Fig.  149,  and  dc 
equal  to  sf,  Fig.  149;  then  draw  do  parallel  to  ci,  and  com- 
plete the  curve  as  before. 

272. — Railing  for  Platform  Stair*  where  the  Rake 
meets  the  Level. — In  Fig.  150,  abc  is  the  plan  of  a  line  pass- 
ing through  the  centre  of  the  rail  around  the  cylinder  as 
before,  and  je  is  a  vertical  section  of  two  steps  starting 
from  the  floor,  kg.  Bisect  eh  in  d,  and  through  d  draw  df 


FIG.  149. 


HAND-RAIL  AT   RAKE   AND   LEVEL. 


273 


parallel  to  hg\  bisect  /#  in  /,  and  from  /  draw  It  parallel 
to  nj\  from  n  draw  nt  at  right  angles  to  jn ;  on  the  line  ob 
make  ot  equal  to  nt.  Then,  to  obtain  a  mould  for  the  twist 
going  up  the  flight,  proceed  as  at  Fig.  145  ;  making  ec  in 
that  figure  equal  to  en  in  Fig.  150,  and  the  other  lines  of  a 
length  and  position  such  as  is  indicated  by  the  letters  of 
reference  in  each  figure.  To  obtain  the  mould  for  the  level 


FIG.  150. 

rail,  extend  bo  (Fig.  150)  to  i ;  make  oi  equal  to  //,  and  join 
z'andc;  vcw&e  c  i  (Fig.  151)  equal  to  civ&Fig.  150;  thiough 


FIG.  151. 

c  draw  cd  at  right  angles  to  ci\  make  dc  equal  to  df  at 
Fig.  150,  and  complete  the  parallelogram  odd;  then  pro- 
ceed as  in  the  previous  cases  to  find  the  mould. 

273.— Application  of  Face-lWoiilds  to  Plank.  — All  the 

moulds  obtained  by  the  preceding  examples  have  been  for 
round  rails.  For  these,  the  mould  may  be  applied  to  a  plank 
of  the  same  thickness  as  the  rail  is  intended  to  be,  and  the 


274 


STAIRS. 


plank  sawed  square  through,  the  joints  being  cut  square 
from  the  face  of  the  plank.  A  twist  thus  cut  and  truly 
rounded  will  hang  in  a  proper  position  over  the  plan,  and 
present  a  perfect  and  graceful  wreath. 

274. — Face-Moulds   for  Moulded  Rails  upon  Platform 
Stairs. — In  Fig.  i$2,abcis  the  plan  of  a  line  passing  through 


FIG.  152. 

the  centre  of  the  rail  around  the  cylinder,  as  before,  and  the 
lines  above  it  are  a  vertical  section  of  steps,  risers,  and  plat- 
form, with  the  lines  for  the  rail  obtained  as  in  Fig.  144.  Set 
half  the  width  of  the  rail  from  b  to  /  and  from  b  to  r,  and 
from  /  and  r  draw  .fc  and  rd  parallel  to  ca.  At  Fig.  153 
the  centre-lines  of  the  rail  jc  and  cl  are  obtained  as  in  the 
previous  examples,  making  jc  equal  jn  of  Fig.  152,  ct 


FACE-MOULD   APPLIED   TO   PLANK.  2/5 

equal  ct  of  Fig.  152,  and  tl  equal  si  of  Fig.  152.  Make  ci 
and  ck  each  equal  to  <:z  at  /^.  152,  and  draw  the  lines  im 
and  /&£•  parallel  to  cj ;  make  /<?  and  /r  equal  to  ne  and  ?z</  at 
7%.  152,  and  draw  dn  and  eq  parallel  to  lc\  also,  through  j 
draw  <?£•  parallel  to  lc;  then,  in  the  parallelograms  mnro 
and  goe  q,  find  the  elliptic  curves,  d??«  and  *£•,  according  to 
Art.  551,  and  they  will  define  the  curves.  The  line  dp, 
being  drawn  through  /  perpendicular  to  Ic,  defines  the 
joint  which  is  to  be  cut  square  through  the  plank. 

275. — Application  of  Face-Moulds  to  Plank. — In  Fig. 
152  make  a  drawing,  from  d  to  /*,  of  the  cross-section  of  the 
hand-rail,  and  tangent  to  the  lower  corner  draw  the  line  gh. 
The  distance  between  the  lines/*  and^/z  is  the  thickness  of 
the  plank  from  which  the  rail  is  to  be  cut.  Lay  the  face1 
mould  upon  the  plank,  mark  its  shape  upon  the  plank,  and 


FIG.   153. 

saw  it  square  through.  To  proceed  strictly  in  accordance 
with  the  requirements  of  the  principles  upon  which  the  face- 
mould  is  formed,  the  cutting  ought  to  be  made  vertically 
through  the  plank,  the  latter  being  in  the  position  which  it 
would  occupy  when  upon  the  stairs.  Formerly  it  was  the 
custom  to  cut  it  thus,  with  its  long  raking  lines.  But,  owing 
to  the  great  labor  and  inconvenience  of  this  method,  efforts 
were  made  to  secure  an  easier  process.  By  investigation  it 
was  found  that  it  was  possible,  without  change  in  the  face- 
mould,  to  cut  the  plank  square  through  and  still  obtain  the 
correct  figure  for  the  railing,  and  this  method  is  the  one  now 
usually  pursued.  Not  only  is  the  labor  of  sawing  much  re- 
duced by  this  change  ;  but  to  the  workman  it  is  an  entire  re- 
lief, as  he  now,  after  marking  the  form  of  the  wreath  upon  the 
plank,  sends  it  to  a  steam  saw-mill,  and,  at  a  small  cost,  has  it 


STAIRS. 


cut  out  with  an  upright  scroll-saw.  When  thus  cut  out  in 
the  square,  the  upper  surface  of  the  plank  is  to  be  faced  up 
true  and  unwinding,  and  the  outer  edge  jointed  straight 
and  square  from  the  face.  Then  a  figure  of  the  cross-section 
of  the  hand-railing  is  to  be  carefully  drawn  on  the  ends  of  the 
squared  block  as  shown  in  Figs.  154  and  155,  and  which 
are  regulated  so  as  to  be  correctly  in  position,  as  follows. 
First,  as  to  the  end  h  of  the  straight  part  hj\  In  Fig.  154, 
let  a  b  c  d  be  an  end  view  of  the  squared  block,  of  which  a  e  fd 
is  the  shape  of  the  end  of  the  straight  part.  Let  the  point  g 
be  the  centre  of  this  end  of  the  straight  part ;  through  g 
draw  upon  the  end  a  efd  the  line  jk,  so  that  the  angle  bjk 
shall  be  equal  to  the  angle  kt  c,  Fig.  152.  This  is  the  angle 
at  which  the  plank  is  required  to  be  canted,  revolving  it  on 


FIG.  154. 


FIG.  155. 


the  axis  of  the  straight  part  of  the  rail.  Through  g  draw 
the  line  ;/  h  parallel  with  a  b.  Upon  a  thin  sheet  of  metal 
(zinc  is  preferable)  mark  carefully  the  exact  figure  of  the 
cross-section  of  the  rail,  drawing  a  vertical  line  through  its 
centre,  cut  away  the  surplus  metal,  then,  with  this  template 
as  a  pattern,  mark  upon  the  end  a  efd,  Fig.  \  54,  the  figure  of 
the  rail  as  shown,  the  vertical  line  upon  the  template  being 
made  to  coincide  with  the  line//£.  From  n  and  h  draw  the 
vertical  lines  //  in  and  /;/  parallel  with  j k. 

Now,  as  to  the  other  end  of  the  square  block  :  Let  b  c  f  e, 
Fig.  155,  represent  the  block,  of  which  bcvn  is  the  form  of 
the  end  at  the  curved  part,  and  o  its  centre.  Through  o 
draiv/^,  so  that  the  angle  epq  shall  be  equal  to  the  angle 
j  n  /7,  Fig.  152.  Also,  through  o  draw  d  h  parallel  with  e  b\ 


CUTTING   THE   TWIST-RAIL.  277 

from  d  and  h  draw  the  vertical  lines  h  r  and  ds  parallel  with 
pq.  Place  the  template  on  bcvn,  the  end  of  the  block,  so 
that  the  vertical  line  through  its  centre  shall  coincide  with 
pq\  mark  its  form,  then  from y,  at  mid-thickness,  draw  wy 
parallel  with  p  q. 

In  applying-  the  mould,  let  Fig.  156  represent  the  upper 
face  of  the  squared  block, 
with  the  face-mould  lying 
upon  it.  With  the  distance 
a  /,  Fig.  154,  and  by  the 
edge  a  x,  mark  a  gauge-line 
upon  the  upper  face  of  the  FIG.  156. 

squared  block.  Set  the  outer  edge  of  the  face-mould  to  coin- 
cide with  this  gauge-line.  Let  the  end  of  the  face-mould  be 
set  at  w,  e  w  being  equal  to  e  w,  Fig.  155;  then  mark  the 
block  by  the  edge  of  the  face-mould. 

Now  turn  the  block  over  and  apply  the  face-mould  to  the 
underside,  as  in  Fig.  157.  With  the  distance  d  r/tt  Fig.  154, 
and  by  the  outer  edge  of 
the  block,  mark  a  gauge- 
line  from  m,  Fig.  157.  Set 
the  inner  edge  of  the  face- 
mould  to  this  gauge-line, 
and  slide  it  endwise  till  the  FlG-  J57. 

distance  em  shall  equal  ew,  Fig.  155,  then  mark  the  block  by 
the  edges  of  the  face-mould.  The  over  wood  may  now  be  re- 
moved as  indicated  by  the  vertical  lines  at  the  sides  of  the 
cross-section  marked  on  each  end  of  the  block  (see  also  Fig. 
167) :  the  direction  of  the  cutting  at  the  curves  must  be  verti- 
cal ;  the  inner  curve  will  require  a  round-faced  plane.  A  com- 
parison of  the  several  figures  referred  to,  with  the  directions 
given,  together  with  a  little  reflection,  will  manifest  the 
reasons  for  the  method  here  given  for  applying  the  face- 
mould.  Especially  so  when  it  is  remembered  that  the  face- 
mould  was  obtained  not  for  the  top  of  the  rail,  but  for  the  rail 
at  the  mid-thickness  of  the  block.  So,  therefore,  in  the 
application  to  the  upper  surface  of  the  block,  the  face-mould 
is  slid  up  the  rake  far  enough  to  put  the  mould  in  position 
vertically  over  its  true  positional  mid-thickness  ;  and  on  the 


2/8  STAIRS. 

contrary,  in  applying  the  face-mould  to  the  underside  of  the 
plank,  it  is  slid  down  until  it  is  vertically  beneath  its  true 
position  at  the  mid-thickness  of  the  block. 

When  the  vertical  faces  are  completed,  the  over  wood 
above  and  below  the  wreath  is  to  be  removed.  In  doing 
this,  the  form  at  the  ends,  as  given  by  the  template,  is  a  suf- 
ficient guide  there.  Between  these  the  upper  and  under 
surfaces  are  to  be  warped  from  one  end  to  the  other,  so  as 
to  form  a  graceful'  curve.  With  a  little  practice  an  intelli- 
gent mechanic  will  be  able  to  work  these  surfaces  with 
facility.  The  form  of  cross-section  produced  by  this  opera- 
tion is  that  of  a  parallelogram,  tangent  to  the  top,  bottom, 
and  two  sides  of  the  rail  ;  and  which  at  and  near  the  ends 
of  the  block  is  not  quite  full.  The  next  operation  is  that  of 
working  the  moulding  at  the  sides  and  on  top,  first  by  re- 
bates at  the  sides,  then  chamfering,  and  finally  moulding  the 
curves.  Templates  to  fit  the  rail,  one  at  the  sides,  another 
on  top,  are  useful  as  checks  against  cutting  away  too  much 
6f  the  wood.  f 

The  joints  are  all  to  be  worked  square  through  the  plank 
in  the  line  drawn  perpendicular  to  the  tangent,  as  shown  in 
153- 


276.  —  Hand-Railing    for  Circular  Stairs.  —  Let  it  be  re- 

quired to  furnish  the  face-moulds  for  a  circular  stairs  similar 
to  that  shown  in  Fig.  133. 

Preliminary  to  making  the  face-moulds  it  is  requisite  to 
make  a  plan,  or  horizontal  projection  of  the  stairs,  and  on 
this  to  locate  the  projections  of  the  tangents  and  develop 
their  vertical  projections.  For  this  purpose  let  b  c  d  e  f  g, 
Fig.  158,  be  the  horizontal  projection  of  the  centre  of  the 
rail,  and  the  lines  numbered  from  i  to  19  be  the  risers.  At 
any  point,  a,  on  an  extension  of  the  line  of  the  first  riser 
locate  the  centre  of  the  newel.  On  a  as  a  centre  describe  the 
two  circles;  the  larger  one  equal  in  diameter  to  the  diame- 
ter of  the  newel-cap,  the  inner  one  distant  from  the  outer 
one  equal  to  half  the  width  of  the  rail.  Let  the  first  joint  in 
the  hand-rail  be  located  at  b,  at  the  fourth  riser  ;  through  b 
draw  h  k  tangent  to  the  circle.  Select  a  point,  h,  on  this 


PLAN   OF  CIRCULAR  STAIRS. 


279 


tangent  which  shall  be  equally  distant  from  b  and  from  the 
inner  circle  of  the  newel-cap,  measured  on  a  line  tending  to 
a  ;  join  h  and  a,  and  from  a  point,  <?,  on  the  line  b  o  describe 


ft, 


10 


FIG.  158. 

the  curve  from  b  to  the  point  of  the  mitre  of  the  newel-cap, 
the  curve  being  tangent,  at  this  point,  to  the  line  a  h.  Select 
positions  for  the  other  joints  in  the  hand-rail  as  at  c,  d,  c,  and  /. 


280  STAIRS. 

Through  these  draw  lines  tangent  to  the  circle.*  Then  the 
horizontal  projection  of  the  tangents  will  be  the  lines  hk,  kl, 
/;//,  m  n,  and  np.  Now,  if  a  vertical  plane  stand  upon  each 
of  these  lines,  these  planes  would  form  a  prism  not  quite 
complete  standing  upon  the  base-plane,  A.  Upon  these  ver- 
tical planes,  C,  D,  E,  F,  G,  and  H,  lines  may  be  drawn  which 
at  each  joint  shall  be  tangent  to  the  central  line  of  the  rail. 
These  are  the  tangents  now  to  be  sought.  Perpendicular  to 
the  tangents  at  £,  c,  d,  etc.,  draw  the  lines  b  bt,  c  c^  ddjy  e  et,  ff/y 
ggn  and  h  htl,  kkt,  k  klt,l  lt,l  lin  etc.  As  b  is  at  the  fourth  riser, 
and  the  height  is  counted  from  the  top  of  the  first  riser, 
make  b  bt  equal  to  three  risers.  (To  avoid  extending  the 
drawing  to  inconvenient  dimensions,  the  heights  in  it  are 
made  only  half  their  actual  size.  As  this  is  done  uniformly 
throughout  the  drawing,  this  reduction  will  lead  to  no  error 
in  the  desired  results.)  As  c  is  on  the  eighth  riser,  therefore 
make  c  ct  equal  to  seven  risers,  and  so,  in  like  manner,  make 
the  heights  ddltee^  and  fft  each  of  a  height  to  correspond 
with  the  number  of  the  riser  at  which  it  is  placed,  deduct- 
ing one  riser.  These  heights  fix  the  location  of  each  tangent 
at  its  point  of  contact  with  the  central  line  of  the  rail.  But 
each  tangent  is  yet  free  to  revolve  on  this  point  of  contact, 
up  or  down,  as  may  be  required  to  bring  the  ends  of  each 
pair  of  tangents  in  contact;  or,  to  make  equal  in  height  the 
edges  of  each  pair  of  vertical  planes,  which  coincide  after 
they  are  revolved  on  their  base-lines  into  a  vertical  position  ; 
as,  for  example  :  the  edges  k  kt  and  k  kit  of  the  planes  C  and 
D  must  be  of  equal  height;  so,  also,  the  edges  llf  and  lltl  of 
the  planes  D  and  E  must  be  of  equal  height.  The  method 
of  establishing  these  heights  will  now  be  shown. 

To  this  end  let  it  be  observed,  that  of  the  horizontal  pro- 
jection of  any  pair  of  intersecting  tangents,  their  lengths, 
from  the  point  of  intersection  to  the  points  of  contact  with 
the  circle,  are  equal ;  for  example :  of  the  two  tangents  //  k 
and  Ik,  the  distances  from  k,  their  point  of  intersection,  to  b 
and  c,  their  points  of  contact  with  the  circle,  are  equal ;  and 
so  also  cl  equals  <//,  dm  equals  e  m,  etc.  It  will  be  observed 

*  A  tangent  is  a  line  perpendicular  to  the  radius,  drawn  from  the  point  of 
contact. 


THE   FALLING-MOULD   FOR   THE   RAIL.  28 1 

that  this  equality  is  not  dependent  on  b,c,  d,  etc.,  the  points 
of  contact,  being  disposed  at  equal  distances ;  for,  in  this 
example,  they  are  placed  at  unequal  distances,  some  being  at 
three  treads  apart  and  others  at  four;  and  yet  while  this  un- 
equal distribution  of  the  points  b,  c,  d,  etc.,  has  the  effect  of 
causing  the  point  of  contact,  as  b,  c,  or  ey  to  divide  each  whole 
tangent  into  two  unequal  parts,  it  does  not  disturb  the 
equality  of  the  two  adjoining  parts  of  any  two  adjacent  tan- 
gents. Now,  because  of  this  equality  of  the  two  adjoining 
parts  of  a  pair  of  tangents,  the  height  to  be  overcome  in 
passing  from  one  point  of  contact  to  the  next  must  be 
divided  equally  between  the  two ;  each  tangent  takes  half 
the  distance.  Therefore,  for  stairs  of  this  kind,  the  arrange- 
ment being  symmetrical,  we  have  this  rule  by  which  to  fix 
the  height  of  the  ends  of  any  two  adjoining  tangents,  namely  : 
To  the  height  at  the  lower  point  of  contact  add  half  the  dif- 
ference between  the  heights  at  the  two  points  of  contact ; 
the  sum  will  be  the  required  height  of  the  two  adjoining 
ends  of  tangents.  For  example:  the  heights  at  b  and  c, 
two  adjacent  points  of  contact,  are  respectively  three  and 
seven  risers;  the  difference  is  four  risers;  half  this  added  to 
three,  the  height  of  the  lower  rise,  gives  five  risers  as  the 
height  of  k  kr  kku,  the  height  at  the  adjoining  ends  of  the 
tangents  h  k  and  /  k.  Again,  the  heights  at  c  and  d  are  re- 
spectively seven  and  ten  risers ;  their  difference  is  three ; 
half  of  which,  or  one  and  a  half 'risers,  added  to  seven,  the 
height  at  the  lower  point  of  contact,  makes  nine  and  a  half 
risers  as  the  heights  //,,  //„,  at  the  ends  of  the  adjoining 
tangents  k  I  and  m  I.  In  a  similar  manner  are  established 
the  heights  of  the  tangents  at  ;;/,  ;/,  and  /. 

The  rule  for  finding  the  heights  of  tangents  as  just  given 
is  applicable  to  circular  stairs  in  which  the  treads  are  di- 
vided equally  at  the  front-string,  as  in  Fig.  158.  Stairs  of 
irregular  plan  require  to  have  drawn  an  elevation  of  the 
rail,  stretched  out  into  a  plane,  upon  which  the  tangents  can 
be  located.  This  will  be  shown  farther  on. 

The  locations  of  the  joints  c,  d,  c,  in  this  example,  were 
disposed  at  unequal  distances  merely  to  show  the  effect  on 
the  tangents  as  before  noticed.  In  practice  it  is  proper  to 


282  STAIRS. 

locate  them  at  equal  distances,  for  then  one  face-mould  in 
such  a  stairs  will  serve  for  each  wreath. 

When  the  tangent  at  G  has  been  drawn,  the  level  tangent 
for  the  landing  maybe  obtained  in  this  manner:  As  the 
joint  f  is  located  at  the  eighteenth  riser,  one  riser  below  the 
landing,  draw  a  horizontal  line  at  s,  one  riser  above  the  point 
f^  and  at  half  a  riser  above  this  draw  the  level  line  at  pt  \  then 
this  line  is  the  level  tangent,  and  p  its  point  of  intersection 
with  the  raking  tangent.  Draw  the  vertical  line  /,/,  and 
from/  draw  the  tangent /£-,  which  is  the  horizontal  projec- 
tion of  the  tangent  pt  g,  on  plane  H  (which,  to  avoid  undue 
enlargement  of  the  drawing,  is  reduced  in  height),  where 
////equals//,,. 

To  obtain  the  horizontal  tangent  /  u  at  the  newel,  pro- 
ceed thus :  Fix  the  point  r,  in  the  tangent  r  kt,  at  a  height 
above  b  t  equal  to  the  elevation  of  the  centre  of  the  newel 
above  the  height  of  a  short  baluster — for  example,  from  5 
to  8  inches — and  draw  a  line  through  r  parallel  to  b  t ;  this 
is  a  horizontal  line  through  the  middle  of  the  height  of  the 
newel-cap,  and  upon  which  and  the  rake  the  easement  to 
the  newel  is  formed.  Perpendicular  to  b  t  draw  r  t,  and  join 
/  and  u  ;  then  /  u  is  the  horizontal  tangent. 

277.— Face-Moulds  for  Circular  §tair§. — At  Fig.  159  the 
plan  of  the  newel  and  the  adjacent  hand-rail  are  repeated, 
but  upon  an  enlarged  scale  ;  and  in  which  b  bt  is  the  reduced 
height  of  the  point  &,  or  is  equal  to  b  bt  less  tr,  Fig.  158, 
and  the  angle  b  bt  /equals  the  angle  bbtr  of  Fig.  158.  In 
this  plan  the  actual  heights  must  now  be  taken.  Join  t  and 
u  ;  then  /  u  is  the  level  tangent,  as  also  the  line  of  intersection 
of  the  cutting  plane  C  and  the  horizontal  plane  A.  Perpen- 
dicular to  /  u,  at  a  point  /  or  anywhere  above  it,  draw  u,  b/r 
Parallel  with  /  u  draw  b  bltl ;  make  bn  bltl  equal  to  b  bt ;  join 
bltl  and  u^ ;  then  the  angle  b  bul  ut  is  the  angle  which  the 
plank  in  position  makes  with  a  vertical  line,  or  what  is 
usually  termed  the  plumb-beviL  Perpendicular  to  bltl  uj 
draw  «,  «„  and  bltl  bltll ;  make  btll  btlll  equal  to  bbtl ;  make  u, 
tt  equal  to  ut  t,  and  uj  un  to  u/  u  ;  join  blltl  and  tt ;  then  bini  tt  is 
the  tangent  in  the  cutting  plane,  the  horizontal  projection  of 


FACE-MOULD    FOR   FIRST   SECTION. 


283 


which  is  bt.  The  butt-joint  at  bltil  is  drawn  square  to  the 
tangent  bilu  tr  Parallel  to  the  intersecting  line  /  u,  draw 
ordinates  across  the  plane  A  from  as  many  points  as  desir- 
able, and  extend  them  to  the  rake-line  ut  bni ;  through  the 
points  of  their  intersection  with  this  line,  and  perpendicular 
to  it,  draw  corresponding  ordinates  across  the  plane  C.  Make 
du  dtll  equal  to  d,  d,  and  so  in  like  manner,  for  all  other  points, 


FIG.  159. 

obtain  in  the  plane  C  for  each  point  in  the  horizontal  plane 
A  its  corresponding  point  in  the  plane  C:  in  each  case 
taking  the  distance  to  the  point  in  the  plane  A  from  the  line 
h  btl  and  applying  it  in  the  plane  C  from  the  rake-line  u,  bnl. 
For  the  curves  bend  a  flexible  strip  to  coincide  with  the 
several  points  obtained,  and  draw  the  curve  by  the  side  of 
the  strip.  The  point  of  the  mitre  is  at  */,„,  the  mitre-joint  is 


284  STAIRS. 

shown  at  hdtll  and  dtllclt.  The  line  f  clt  is  drawn  through 
c//t  the  most  projecting  point  of  the  mitre,  and  parallel  to  the 
rake-line  ut  blir  Additional  wood  is  left  attached,  extending 
from  h  to  f\  this  is  an  allowance  to  cover  the  mitre,  which 
has  to  be  cut  vertically ;  the  butt-joint  at  bltll  and  the  face  at 
fctl  are  both  to  be  cut  square  through  the  plank.  The  face 
/<:„,  because  it  is  parallel  to  the  rake-line  uf  bltl,  is  a  vertical 
face,  as  well  as  being  perpendicular  to  the  surface  of  the 
plank.  On  it,  therefore,  lines  drawn  according  to  the  rake, 
or  like  the  angle  ut  btll  bit,  will  be  vertical  and  will  give  the 
direction  of  the  mitre-faces.  We  now  have  at  C  the  face- 
mould  for  the  railing  over  the  plan  from  b  to  d  in  A.  The 
mould  thus  found  is  that  made  upon  a  cutting  plane  C,  passed 
through  the  plank,  parallel  to  its  face,  but  at  the  middle  of 
its  thickness.  To  put  it  in  position,  let  the  plane  C  be  lifted 
by  its  upper  edge  ctl  and  revolved  upon  the  line  nt  btll  until 
it  stands  perpendicular  to  the  plane  B.  Now  revolve  both 
C  and  B  (kept  in  this  relative  position  during  the  revolution) 
upon  the  line  ut  bt.  until  the  plane  B  stands  perpendicular  to 
the  plane  A.  Then  every  point  upon  plane  C  will  be  verti- 
cally over  its  corresponding  point  in  the  plane  A.  For  ex- 
ample, the  point  blllt  will  be  vertically  over  b,  t,  over  /, 
and  so  of  all  other  points.  To  show  the  application  of  the 
face-mould  to  the  plank,  make  bltl  bv  equal  to  half  the  thick- 
ness of  the  plank;  parallel  to  ut  btll  draw  bvc,  a  line  which 
represents  the  upper  surface  of  the  plank,  for  the  line  ut  bul 
is  at  the  middle  of  the  thickness.  Through  bitll,  and  parallel 
with  bltl  ut,  draw  the  line  ct  bltll  and  extend  it  across  the  face- 
mould  ;  make  blitl  ct  equal  to  bv  c ;  through  c^  and  parallel  with 
bull  t/t  draw  ct  e.  Now,  ;//  n  ot  p  is  an  end  view  of  the  plank, 
showing  the  face  view  of  the  butt-joint  at  bi{ll.  Through  r, 
the  centre,  draw  a  line  parallel  with  the  sides.  Then  #vl  rep- 
resents the  point  bltll\  make  £vi  et  equal  to  btlil  c\  through  r, 
the  centre,  draw  c,  r  across  the  face  of  the  joint ;  then  et  r  is 
a  vertical  line  (see  Art.  284),  parallel  and  perpendicular  to 
which  the  four  sides  of  the  squared-up  wreath  are  to  be 
drawn  as  shown.  In  applying  the  face-mould  to  the  plank  at 
first,  for  the  purpose  of  marking  by  its  edges  the  form  of  the 
face-mould,  it  will  be  observed  that  the  face-mould  is  under- 
stood to  have  the  position  indicated  by  the  line  ut  blu,  or  at 


FACE-MOULDS   FOR   CIRCULAR   STAIRS.  285 

the  middle  of  the  thickness  of  the  plank.  By  this  marking 
the  rail-piece  is  cut  square  through  the  plank,  and  this  cut- 
ting gives  the  correct  form  of  the  wreath,  but  only  at  the 
middle  of  the  thickness  of  the  plank.  After  it  is  cut  square 
through  the  plank,  then,  to  obtain  the  form  at  the  upper  and 
under  surfaces,  the  face-mould  is  required  to  be  moved  end- 
wise, but  parallel  with  the  auxiliary  plane  ^,-and  so  far  as  to 
bring  the  face-mould  into  a  position  vertically  over  or  under 
its  true  position  at  the  middle  of  the  thickness  of  the  plank. 
For  example,  the  point  btlll^  if  the  mould  were  placed  at  the 
middle  of  the  thickness  of  the  plank,  would  be  at  the  height 
of  the  point  bnl;  but  when  upon  the  top  of  the  plank,  the 
point  btlll  would  have  to  be  at  the  height  of  the  point  ct, 
therefore  the  mould  must  be  so  moved  that  the  point  blltl 
shall  pass  from  bv  to  c ;  consequently  bv  c  is  the  distance 
the  mould  must  be  moved,  or,  as  it  is  technically  termed, 
the  sliding  distance;  hence  blltl  c,,  which  is  equal  to  bv  c,  is 
the  distance  the  mould  is  to  be  moved:  up  when  on  top, 
and  down  when  underneath.  This  is  more  fully  explained 
in  Art.  284. 

278. — Face-Moulds  for  Circular  Stairs. — At  Fig.  1 60  so 
much  of  the  horizontal  projection  of  the  hand-railing  of 
stairs  in  Fig.  158  is  repeated  as  extends  from  the  joint  b  to 
that  at  d,  but  at  an  enlarged  scale.  Upon  the  tangent  ck 
set  up  the  heights  as  given  in  Fig.  158;  for  example,  make 
kkt  equal  to  kniktl  of  Fig.  158,  and  cc,  equal  to  cllcl  of  Fig. 
158.  Join  ct  and  kt  and  extend  the  line  to  meet  ck,  extended, 
in  a.  Join  a  and  b  ;  then  ab  is  the  line  of  intersection  of 
the  cutting  and  horizontal  planes ;  it  is  therefore  a  horizon- 
tal line,  parallel  to  which  the  ordinates  are  to  be  drawn. 
Perpendicular  to  ab  draw  b,c,til.  Parallel  to  ab  draw  ccu 
and  kklt ;  join  b,  and  clt ;  the  angle  cctl  bf  is  the  plumb-bevil ; 
perpendicular  to  b,  cf.  draw  b,  blt,  ktl  kin,  and  ctl  cllt ;  make  bt  bu 
equal  to  bt  b,  and  so  of  the  other  two  points,  kltl  and  r///}  make 
them  respectively  equal  to  their  horizontal  projections  upon 
the  plane  A.  Join  ciu  and  kfl ;  also,  kit  and  btl ;  then  btlkltl 
and  ktll  cltl  are  the  tangents.  From  t//f  draw  the  line  cltl  blt 
parallel  to  btctl ;  this  is  the  slide-line.  In  this  example,  this 


286 


STAIRS. 


line  passes  through  the  point  blt ;  the  slide-line  does  not 
always  pass  through  the  ends  of  the  two  tangents ;  it  is  not 
required  to  pass  through  both,  but  it  is  indispensable  that  it 
be  drawn  parallel  with  the  rake-line  bt  ctl.  The  lines  for  the 
joints  at  each  end  are  drawn  square  to  the  tangent  lines. 
Points  in  the  curves,  as  many  as  are  desirable,  are  now  to 
be  found  by  ordinates  as  shown  in  the  figure,  and  as  before 


explained  for  the  points  in  the  tangents.  The  curves  are 
made  by  drawing  a  line  against  the  side  of  a  flexible  strip 
bent  to  coincide  with  the  points. 

The  face-mould  may  be  put  in  position  by  revolving  the 
planes  C  and  B,  as  explained  in  the  last  article,  for  the  rail 
at  the  newel. 

The  face-mould  for  the  rail  over  the  plan  from  c  to  d  is  to 


FACE-MOULDS   CONTINUED.  287 

be  obtained  in  a  similar  manner,  taking  the  heights  from  Fig. 
158.  For  example,  make  d  d/  equal  to  dtldt  of  Fig.  15.8,  and 
//,  equal  to  /„,/„  of  Fig.  158  (taking  the  heights  at  their 
actual  measurement  now).  Join  dt  and  /„  and  extend  the 
line  to  meet  the  line  dl  extended  in  r ;  join  r  and  c\  then  re 
is  the  line  of  intersection,  and  parallel  to  which  the  ordinates 
are  to  be  drawn.  The  points  in  the  face-mould  may  now  be 
obtained  as  in  the  previous  cases,  giving  attention  first  to 
the  tangent  and  slide-line ;  drawing  the  lines  for  the  joints 
perpendicular  to  the  tangents. 

It  may  be  remarked  here  that  the  chord-line  £<:is  parallel 
with  the  measuring  line  btcilin  and  that  the  line  o k  bisects 
the  chord-line;  so,  also,  the  line  ol  bisects  the  chord-line  cd. 
This  coincidence  is  not  accidental ;  it  will  always  occur  in  a 
regular  circular  stairs. 

Hence  in  cases  of  this  kind  it  is  not  necessary  to  go 
through  the  preliminaries  by  which  to  obtain  the  intersect- 
ing line  ab,  but  draw  it  at  once  parallel  to  the  line  ok, 
bisecting  the  chord  be  and  passing  through  the  point  of 
intersection  of  the  two  tangents!  For  the  distance  to  slide 
the  mould  in  its  after-application,  the  lines  are  given  at  ctl 
and  dlfj  and  their  use  is  explained  in  the  last  article,  and 
more  fully  in  Art.  284. 

279. — Face-Mould§  for  Circular  Stairs  again. — At  Fig. 
161  so  much  of  the  plan  of  the  hand-railing  of  the  stairs  of 
Fig.  158  is  repeated  as  is  required  to  show  the  rail  from  f 
to  gy  but  drawn  at  a  larger  scale.  To  prepare  for  the  face- 
moulds,  perpendicular  to  //  draw  //.,  and  make  ppt  equal 
t°  PuiPii  °f  F*8-  !5^  (taking  this  height  now  at  its  actual 
measurement) ;  join  pt  and  /;  then  fpt  is  the  tangent  of  the 
vertical  plane  C,  and  /  is  a  point  in  the  cutting  plane  at  its 
intersection  with  the  bass-plane  A.  Now  since  rs,  the  tan- 
gent over  pg,  is  horizontal  and  is  in  the  cutting  plane, 
therefore  from  /  draw  fa  parallel  with  r  s  or  pg\  then  fa 
is  the  line  of  intersection  of  the  cutting  and  horizontal 
planes,  and  gives  direction  to  the  ordinates.  Draw  //,„ 
perpendicular  to  fa  ;  make  plllpjl  equal  to  ppt ;  join  /„  and 
/,;  then  the  angle  //„//  is  the  plumb-bevil ;  perpendicular 


288 


STAIRS. 


to  /„/,  draw  //„  and  /„/„„ ;  make  /„/„„  equal  to  ptug, 
pitd  equal  to  plltp\  join  d  and  /„ ;  then  *//„  and  dplin  are 
the  tangents.  Make  ptl  e  equal  to  half  the  thickness  of  the 
plank ;  draw  ftl  a  parallel  Avith  ftplt ;  make  ftl  a  equal  to  e  c ; 
draw  act  parallel  with  the  tangent  fnd\  through  /„,  per- 
pendicular to  fti  d,  draw  the  line  for  the  butt-joint ;  then  fu  c, 
is  the  distance  required  to  determine  the  vertical  line  on  the 
face  of  the  joint  at  /„,  as  shown  at  A.  Through  pltli,  per- 


pendicular to  the  tangent  pull  d,  draw  the  line  for  the  butt- 
joint ;  make  pnil  b  equal  to  ec\  then  plltlb  is  the  distance 
required  for  determining  the  vertical  line  on  the  face  of  the 
joint  at  ///7/,  as  shown  at  B  (see  Art.  284).  The  curved  lines 
are  obtained  by  drawing  a  line  against,  the  edges  of  a  flexi- 
ble rod  bent  to  as  many  points  as  desirable,  obtained  by 
measuring  the  ordinates  of  the  plan  at  A  and  transferring 
them  to  the  face-mould  by  the  corresponding  ordinates,  as 
before  explained. 


RAILING   FOR  QUARTER-CIRCLE   STAIRS. 


289 


280.—  Hand -Railing  for  Winding  Stairs.— -The  term 
winding  is  applied  more  particularly  to  a  stairs  having  steps 
of  parallel  width  compounded  with  those  which  taper  in 
width,  as  in  Fig.  135,  and  as  is  here  shown  in  Fig.  162,  in 
which  fa  be  represents  the  central  line  of  the  rail  around  the 
cylinder,  and  the  quadrant  de,  distant  from  the  first  quadrant 
20  inches,  is  the  tread-line,  upon  which  from  d,  a  point  taken 
at  pleasure,  the  treads  are  run  off.  Through  e,  perpendicu- 


FIG.   162. 


lar  to  af,  draw  ae  (the  occurrence  here  of  one  of  the 
points  of  division  on  the  tread-line  perpendicularly  opposite 
a,  the  spring  of  the  circle,  is  only  an  accidental  coincidence)  ; 
make  a  a,  equal  to  two  risers ;  join  a,  and  /.  With  the 
diameter  a  c,  on  b  as  a  centre,  describe  the  arc  at  g,  crossing 
ac  extended  ;  through  b  draw  gb, ;  then  ab,  is  the  stretch- 
out, or  development  of  the  quadrant  a  b. 

Through  //  draw  h  i,  tending  toward  the  centre  of  the 


STAIRS. 

cylinder;  make  btit  equal  to  bi\  perpendicular  to  fbt  draw 
bt  bit  and  it  itl.  As  there  are  four  risers  from  e  to  //,  make 
at  ati  equal  to  four  risers,  and  draw  atl  iu  parallel  with  fa  ; 
through  iit  draw  at  bi(  ;  by  intersecting  lines,  or  in  any  con- 
venient manner,  ease  off  to  any  extent  the  angle  fat  iu. 
Through  j,  a  point  in  this  curve  (chosen  so  as  to  be  perpen- 
dicularly over  m,  a  point  between  a  and  /,  nearer  to  a), 
draw  kly  a  tangent  to  the  curve.  Perpendicularly  to  this 
tangent,  through  /,  draw  the  line  for  a  butt-joint  ;  also 
through  £//t  and  perpendicularly  to  a,  blt,  draw  the  line  for 
the  joint  at  the  centre  of  the  half  circle.  On  the  line  aan 
set  up  points  of  division  for  the  riser  heights,  and  through 
these  points  of  division  draw  horizontal  lines  to  the  line 


From  these  points  of  contact  drop  perpendiculars  to  the 
line  fa  bt,  and  transfer  such  of  them  as  require  it  to  the  circle 
at,  by  drawing  lines  tending  to  g.  Through  these  points  of 
intersection  with  the  central  line  of  the  rail,  and  through  the 
points  of  division  on  the  tread-line,  draw  the  riser-lines  me, 
a  n,  etc.  At  half  a  riser  above  the  floor-line,  on  top  of  the 
upper  riser  draw  a  horizontal  line,  and  ease  off  the  angle  as 
shown  ;  the  intersection  of  the  floor-line  with  this  curve 
gives  the  position  of  the  top  riser  at  the  centre  of  the  rail. 
This  completes  the  plan  of  the  steps  and  the  elevation  of  the 
rail  —  requisite  preliminaries  for  the  face-moulds.  The  gradu- 
ation of  the  treads  from  flyers  to  winders  obviates  an  abrupt 
angle  at  their  junction  in  the  rail  and  front-string.  The 
objection  to  the  graduation,  that  it  interferes  with  the 
regularity  of  stepping  at  the  tread-line,  is  not  realized  in 
practice. 

281.  —  Face-Moulds  for  Winding  Stairs.  —  At  Fig.  163  so 
much  of  the  plan  at  Fig.  162  is  repeated  as  is  required  for  the 
face-moulds,  but  for  perspicuity  at  twice  the  size.  The  hori- 
zontal projection  of  the  tangents  for  the  first  wreath  are  ad 
and  db  drawn  at  right  angles  to  each  other,  tangent  to  the 
circle  at  a  and  b.  Let  those  tangents  be  extended  beyond 
d;  through  ;;/,  the  lower  end  of  the  wreath,  draw  mdjy  mak- 
ing an  angle  with  md  equal  to  that  in  Fig.  162,  between  the 


FACE-MOULDS   FOR  THE  TWISTS. 


29I 


line  af  and  #,./;  or  let  the  angle  dmdt  equal  a  fa,  of  Fig. 
162.  Make  ddlt  equal  to  ddr  Make  bbtl  equal  to  btllbtl  of 
Fig.  162  ;  join  dlt  and  btl  and  extend  the  line  to  e4l ;  make 
blt  bv  equal  to  bubllu  of  Fig.  162,  and  draw  b^etl  parallel  with 


FIG.  163. 

</*.  From  ^/y  draw  etle  parallel  with  bltb\  through  e  and  / 
draw  ef  tangent  to  the  circle  at  /;  then  b  e  and  ef  are  the 
horizontal  projections  of  the  tangents  for  the  upper  wreath. 
Then  if  the  plane  B  be  revolved  on  ad,  the  plane  C  on  de, 


2Q2  STAIRS. 

and  the  plane  D  on  cf  until  they  each  stand  vertical  to  the 
plane  A,  the  lines  mdn  dtlein  and  enift  will  constitute  the 
tangents  of  the  two  wreaths  in  position.  This  arrangement 
locates  the  upper  joint  of  the  upper  wreath  at  /,  leaving  fc, 
a  part  of  the  circle,  to  be  worked  as  a  part  of  the  long  level 
rail  on  the  landing.  As  the  tangent  over  ef  is  level,  the 
raking  part  of  the  rail  will  all  be  included  in  the  wreath  bf, 
so  that  at  the  joint  /  the  rail  terminates  on  the  level. 

The  portion  fc,  therefore,  is  a  level  rail  requiring  no 
canting,  and  it  requires  no  other  face-mould  than  that  afforded 
by  the  plan  from  /  to  c. 

For  the  face-mould  for  the  rail  over  ;;/  a  b,  let  the  line  eff  dtl 
be  extended  to  mv,  a  point  in  the  base-line  b  mv ;  then  ;//v  is  a 
point  in  the  base-plane  A,  as  well  as  in  the  cutting  plane  E; 
therefore  the  line  mv  m  is  the  intersecting  line  parallel  to 
which  all  the  ordinates  on  plane  A  are  to  be  drawn.  Per- 
pendicular to  this  intersecting  line  mvm,at  any  convenient 
place  draw  m,  b, ;  make  bt  bni  parallel  to  mv  m  and  equal  to 
b  btl ;  connect  bilt  with  ;//,,  a  point  at  the  intersection  of  the 
lines  mv  m  and  b,  mt ;  then  the  angle  b  bnl  m/  is  the  plumb- 
bevil.  Through  d,  parallel  to  mv  m,  draw^^;  from  the 
three  points  /#,,  diti,  and  btil  draw  lines  perpendicular  to 
mf  bllt ;  make  mt  mtl  equal  to  mt  m  ;  make  bitl  bllu  equal  to  bt  b. 
Since  the  measuring  base-line  m,  b,  passes  through  d,  the 
point  of  the  angle  formed  by  the  two  tangents,  dtil  is  the 
point  of  this  angle  in  the  cutting  plane  E\  therefore  join  mtl 
and  </,„,  also  dltl  and  blltl ;  then  bltll  dni  and  dltl  mn  are  .the 
two  tangents  at  right  angles  to  which  the  joints  at  mn  and 
blin  are  drawn.  The  curves  of  the  face-mould  are  now  found 
as  usual,  by  transferring  the  distances  by  ordinates,  as  shown, 
from  the  plane  A  to  the  plane  E,  making  the  distance  from 
the  rake-line  m/  bltl  to  each  point  in  plane  E  equal  to  the  dis- 
tance from  the  corresponding  point  in  the  plane  A  to  the 
measuring  base-line  m,  bt.  Now,  to  obtain  the  sliding  distance 
and  the  vertical  line  upon  the  butt-joints,  make  btll  bv  equal 
to  half  the  thickness  of  the  plank ;  parallel  with  mt  btll  draw 
by  b^ ;  also,  bllu  £vii  and  m4l  mtll ;  make  btlll  bvYi  and  «*„  mllt 
each  equal  to  bv  £vi ;  through  ^vii  and  mn/,  and  parallel  to  the 
respective  tangents,  draw  £vil  b^  and  mllt  mliu ;  then  b*  and 


FACE-MOULDS   FOR  WINDING  STAIRS.  293 

m,ni  are  the  points  from  which,  through  the  centre  of  the 
butt-joints,  a  line  is  to  be  drawn  which  will  be  vertical  when 
the  wreath  is  in  position.  (See  Art.  284.) 

For  the  face-mould  for  the  upper  quarter,  through  b,  Fig. 

163,  draw  b  et  parallel  with  du  etl ;  make  e  elit  equal  to  e  et ; 
draw  elilfl  parallel  with  e  f.     Now,  since  etll  ft  is  a  horizon- 
tal line  and  is  in  the  cutting  plane  F,  therefore,  parallel  with 
etilft  and  through  btJ  draw  b  n ;  then  b  n  is  the  required  in- 
tersecting line.     Extend  e  f  to  /„ ;  make  //„  equal  to  //, ; 
join/,,  and  n  ;  then  the  angle  ffl{  n  is  the  plumb-bevil.    Per- 
pendicular to  nft/  draw /"„/"„,  and  n  n,,  and  make  these  lines 
respectively  equal  to  e  f  and  b  n  ;  join  ftl  and  fltl ;  also  /„, 
and  nt ;   then  fn  ftil  and  fnl  n{   are  the  required  tangents. 
The  butt-joints  at/)  and  nt  are  drawn  perpendicular  to  their 
respective  tangents.     To  get  the  slide  distance  and  vertical 
lines  on  the  butt-joints,  make/),  fv  equal  to  half  the  thickness 
of  the  plank  ;  parallel  with  n  /),,  through  /v  draw/v/)yy/ ;  also, 
through  nt  draw  nt  nti ;  make  n,  ii'u  equal  to  /v  fntl ;  through 
»/;,  parallel  with  ntfltlJ  draw  nu  nlt.\  then  nln  is  the   point 
through  which  a  line  is  to  be  drawn  to  the  centre  of  the 
butt-joint,  and  this  line  will  be  in  the  vertical  plane  contain- 
ing the  tangent.     So,  also,  parallel  with  the  tangent  ftl  fltl, 
and  through  ////y,  draw  flllt  /vi ;  then/vi  is  the  point  through 
which  a  line  is  to  be  drawn  to  the  centre  of  the  butt-joint 
(see  Art.  284).     The  curve  is  now  to  be   obtained  by  the 
ordinates,  as  before  explained. 

282. — Face-]Woiilci§  for  Winding  Stairs,  again.  — In  the 

last  article,  in  getting  the  face-moulds  for  a  winding  stairs, 
the  two  wreaths  are  found  to  be  very  dissimilar  in.  length. 
This  dissimilarity  may  be  obviated  by  a  judicious  location  of 
the  butt-joint  connecting  the  two  wreaths,  as  shown  in  Fig. 

164.  Instead  of  locating  the  joint  precisely  at  the  middle  of 
the  half  circle,  as  was  done  in  Fig.  163,  place  it  farther  down, 
say  at  ;/,  which  is  at  n  in  Fig.  162,  two  risers  down  from  the 
top,  or  at  any  other  point  at  will.     Then  through  n  in  the 
plan  draw  mt  s  tangent  to  the  circle  at  n ;  and  perpendicu- 
lar to  this  tangent  draw  ;/  nin  and  ddn  ;  make  n  ntl  equal  to 
nt  n  of  Fig.   162;  from  d  erect  d  dt   perpendicular  to  m  d\ 


294 


STAIRS. 


make  the  angle  d  m  dt  equal  to  that  of  bin  j  I  of  Fig.  162. 
Make  d  dn  equal  to  d  dt ;  join  dn  and  nn  and  extend  the  line 
to  ;;/,,  a  point  of  intersection  with  the  base-line  n  nt ;  then  nt 
is  a  point  in  the  base-plane,  as  also  in  the  cutting  plane ; 


FIG.  164. 


therefore  mt  m  is  the  intersecting  line  parallel  to  which  all 
the  ordinates  of  the  plan  are  to  be  drawn,  and  perpendicular 
to  which  m/{  nt,  the  measuring  base-line,  is  drawn.  Make 
nt  ntlil  equal  to  n  nn  ;  connect  mtl  and  n/(/i,  and  then  transfer 


CARE   REQUIRED   IN  DRAWING.  295 

by  the  ordinates  to  the  cutting  plane  m  d  and  n  the  three 
points  of  the  plan  at  the  ends  of  the  tangents,  as  before  de- 
scribed, as  also  such  points  in  the  curve  as  may  be  required 
to  mark  the  curve  upon  the  face-mould,  all  as  shown  in  previ- 
ous examples.  For  the  face-mould  of  the  upper  wreath,  make 
ntl  nnl  equal  to  n  nn  of  Fig.  162.  From  ntll  draw  nnl  stl  par- 
allel  with  mt  s  ;  extend  the  line  dlt  nlt  to  intersect  ntll  su  in  slt ; 
parallel  with  niu  n  draw  s/f  s  ;  from  s  draw  s  r  tangent  to  the 
circle  at  r  (s  n  equals  s  r) ;  through  r,  tending  to  the  centre  of 
the  cylinder,  draw  the  butt-joint ;  then  r  s  and  s  n  are  the 
horizontal  projections  of  the  tangents  for  the  upper  wreath- 
piece,  the  tangent  s  r  being  level  and,  consequently,  parallel 
to  the  intersecting  line  drawn  through  n.  Perpendicular  to 
r  s  draw  rtp  ;  parallel  with  nu  sit  draw  nst ;  make  rt  rlt  equal 
toss,;  join  rn  and /.  From  this  line  and  the  measuring  base- 
line r,  py  the  points  for  the  tangents  are  first  to  be  obtained 
and  then  the  points  in  the  curve,  all  as  before  described. 
The  part  of  the  circle  from  r  to  c  is  on  the  level,  as  before 
shown,  and  may  be  worked  upon  the  end  of  the  long  level 
rail,  its  form  being  just  what  is  shown  in  the  plan  from  c  to  r. 

283. — Face-Moiild§ :  Te§t  of  Accuracy. —  The  methods 
which  have  been  advanced  for  obtaining  face-moulds  are 
based  upon  principles  of  such  undoubted  correctness  that 
there  can  be  no  question  as  to  the  results,  when  the  methods 
given  are  thoroughly  followed.  And  yet,  notwithstanding 
the  correctness  of  the  system  and  its  thorough  comprehen- 
sion by  the  stair-builder,  he  will  fail  of  success  unless  he 
exercises  the  greatest  care  in  getting  his  dimensions,  his  per- 
pendiculars, and  his  angles.  The  slightest  deviation  in  a 
perpendicular  terminated  by  an  oblique  line  will  result  in  a 
magnified  error  at  the  oblique  line.  To  secure  the  greatest 
possible  degree  of  accuracy,  care  must  be  exercised  in  the 
choice  of  the  instruments  by  which  the  drawings  are  to  be 
made  :  care  to  know  that  a  straight-edge  is  what  it  purports 
to  be ;  that  a  square,  or  right-angle,  is  truly  a  right-angle ; 
that  the  compasses  or  dividers  be  well  made,  the  joint  per- 
fect, and  the  ends  neatly  ground  to  a  point.  Then  let  the 
drawing-board  be  carefully  planed  to  a  true  surface  ;  and, 


296  STAIRS. 

if  possible,  let  the  drawing,  full  size,  be  made  upon  large, 
stout  roll-paper  rather  than  upon  the  drawing-board  itself, 
as  then  the  points  for  the  face-mould  may  be  pricked  through 
upon  the  board  out  of  which  the  face-mould  is  to  be  cut,  and 
thus  a  correct  transfer  be  made.  For  long  straight  lines  it 
is  better  to  use  a  fine  chalk-line  than  the  edge  of  a  wooden 
straight-edge.  The  line  is  more  trustworthy.  Perpendicu- 
lars, especially  when  long,  are  better  obtained  by  measure- 
ment or  by  calculation  (Art.  503)  than  by  a  square.  The 
pencil  used  should  be  of  fine  quality — rather  hard,  in  order 
that  its  point  may  be  kept  fine.  With  these  precautions  in 
regard  to  the  instruments  used,  and  with  due  care  in  the 
manipulations,  the  face-moulds  may  be  correctly  drawn, 
accurate  in  size  and  form.  As  a  test  of  the  accuracy  of  the 
work,  it  will  be  well  to  observe  in  regard  to  the  tangents, 
that  the  length  of  a  tangent,  as  found  upon  the  face-mould, 
should  always  equal  its  length  as  shown  upon  the  vertical 
plane.  For  example,  in  Fig.  160,  the  tangent  klt  ctil  on  the 
face-mould  should  be  equal  to  kt  ct,  the  tangent  on  the  vertical 
plane  B ;  and  in  cases  like  this,  where  the  stairs  are  quite 
regular,  with  equal  treads  at  the  front-string,  the  two  tan- 
gents of  a  face-mould  are-equal  to  each  other,  or  klf  cin  equals 
ktl  btl ;  and  in  this  case,  the  line  btl  cllt  should  equal  the  rake- 
line  b,  *„. 

Again,  as  another  example,  in  Fig.  161,  d  fin  the  tangent 
upon  the  face-mould,  should  be  equal  to//,,  the  tangent  of 
the  vertical  plane  C\  while  d  ///;/,  the  other  tangent  on  the 
face-mould,  should  be  equal  to  r  s,  the  tangent  of  the  vertical 
plane  D.  But  the  more  important  test  is  in  the  length  of  the 
chord-line  joining  the  ends  of  the  two  tangents  ;  as,  for  ex- 
ample, the  chord  mtl  binl  of  Fig.  163,  the  horizontal  projec- 
tion of  which  is  the  chord  ;;/  b  in  plane  A.  Perpendicular  to 
m  b  draw  b g\  make  b g  equal  to  b  blt,  and  join  g  and  m  ;  then 
m^  blilt,  the  chord  of  the  face-mould,  should  be  equal  to  ;//  g. 
After  fully  testing  the  accuracy  of  the  drawing  for  the  face- 
mould,  choose  a  well-seasoned  thin  piece  of  white-wood,  or 
any  other  wood  not  liable  to  split,  and  plane  it  to  an  even 
thickness  throughout ;  mark  upon  it  the  curves,  joints,  tan- 
gents, and  slide-line,  and  cut  the  edges  true  to  the  curve- 


THE   FACE-MOULD   APPLIED   TO   PLANK.  297 

lines  and  joints  square  through  the  board ;  then  square  over 
such  marks  as  are  required  to  draw  each  tangent  and  the 
slide-line  also  upon  the  reverse  side  of  the  board.  This 
completes  the  face-mould. 

284. — Application  of  the  Faee-UIoulcl. — In  order  that  a 
more  comprehensive  idea  of  the  lines  given  for  applying  a 
face-mould  may  be  had,  let  A,  Fig.  165,  represent  one  end  of 
a  wreath-piece  as  it  appears  when  first  cut  from  a  plank,  and 
when  held  up  in  the  position  it  is  to  occupy  at  completion 
over  the  stairs.     Also,  let  B  represent  the   corresponding 
face-mould,  laid  upon  the  wreath-piece  A  in   the  position 
which  it  should  have  after  sliding.     And,  for  the  purpose  of 
a  clearer  illustration,  let  it  be  supposed  that  the  two  pieces, 
A  and  B,  are  transparent.     Then  let  at  a  b  d  c^  et  represent  a 
solid  of  wedge  form,  having  a  triangular  level  base,  a  b  d, 
upon  the  three  lines  of   which  stand  these  three  vertical 
planes,  namely :  on  the  line  a  b  the  plane  at  a  b  c^  upon  the 
line  a  d  the  plane  a,  a  d  et,  and  on  the  line  d  b  the  plane  b  d 
et  c, ;  the  top  of  the  solid  is  an  inclined  plane,  ai  ct  en  and  the 
vertical  line  at  a  is  the  edge  of  the  wedge.     Now,  it  will  be 
observed  that  the  point  a  in  the  base  of  the  solid  is  identical 
with  a,  the  centre  of  the  butt-joint,  and  the  point  a,  (at  the 
intersection  of  two  vertical  planes  and  the  inclined  plane  of 
the  solid)  is  vertically  over  a,  and  is  identical  with  a,,  a  poim 
in  the  upper  surface  of  the  plank.     Also,  the  inclined  plane 
e/  c/  at,  which  forms  the  top  of  the  solid,  coincides  with  the 
upper  surface  of  the  plank  A,  from  which  the  wreath-piece 
has  been  squared  ;  and  the  line  c,  at  (at  the  angle  formed  by 
the  inclined  plane  e,  c,  a,  and  the  vertical  plane  af  a  b  c,)  coin- 
cides with  /  g,  the  slide-line  drawn  upon  the  top   of  the 
plank ;  also,  the   line  e,  a,  (at  the  angle  formed  by  the  in- 
clined plane  e,  c,  at  and  the  vertical  plane  at  a  d  e^  coincides 
with  at  k,  the  tangent  line  upon  the  underside  of  the  face- 
mould  after  it  has  been  slid   to  its  new  position,  vertically 
over  its  true  position  at  the  middle  of  the  thickness  of  the 
plank.     From  a  the  line  a  c  is  drawn  parallel  with  a,  ct ;  so, 
also,  the  line  a  e  is  drawn  parallel  with  a,  e, ;  consequently 
the  line  e  c  is  parallel  with  e,  ct ;  and  the  plane  e  c  a  is  parallel 


298 


STAIRS. 


with  the  plane  e{  cf  at,  and  coincides  with  a  plane  passing 
through  the  middle  of  the  thickness  of  the  plank,  and,  conse- 
quently, is  the  cutting  plane  referred  to  in  previous  articles, 
upon  which  the  lines  are  drawn  which  give  shape  to  the 


FIG.  165. 


face-mould.  When  the  face-mould  is  first  laid  upon  the  plank, 
the  line  i,j,  coincides  with  ilt  j/t,  and  Hvhen  in  that  position, 
its  form  marked  upon  the  plank  is  the  form  by  which  the 
plank  is  sawed  square  through  ;  but  this  gives  the  form  of 


THE   SLIDING  OF  THE   FACE-MOULD.  299 

the  wreath,  not  as  it  is  at  the  surface  of  the  plank,  but  as  it 
is  at  the  middle  of  the  thickness  of  the  plank,  or  upon  the 
plane  ace-,  so  that,  for  example,  the  line  iitjtl  represents  the 
line  ij  drawn  through  a,  the  centre  of  the  butt-joint ;  and 
when  the  mould  B  is  slid  to  the  position  shown  in  the  figure, 
the  line  iljl  comes  into  a  position  vertically  over  ij\  hence 
the  three  lines  if  i,  at  a,  and  jtj  are  each  vertical  and  in  a 
vertical  plane,  Hjj^j-  By  these  considerations  it  will  be  seen 
that  the  face-mould  B,  located  as  shown  in  the  figure,  is  in  its 
true  position  for  the  second  marking,  by  which  the  addi- 
tional cutting  is  now  to  be  performed  vertically.  This  being 
established,  it  will  now  be  shown  how  to  get  upon  the  butt- 
joint  a  line  in  the  vertical  plane  containing  the  tangent.  If 
the  top  and  bottom  lines  of  the  vertical  plane  at  a  b  ct  be  ex- 
tended, they  will  meet  in  the  point  /,  and  will  extend  the 
plane  into  a  triangle  Ib  ct,  cutting  the  upper  edge  of  the 
butt-joint  in/,  the  end  of  the  tangent,  and  the  point  in  which 
the  point  at  of  the  underside  of  the  face-mould  was  located 
when  the  mould  was  first  applied  to  the  plank.  The  line/Vz 
on  the  butt-joint  is  perpendicular  to  i  j  or  ilt  ju.  Again,  if 
the  top  and  bottom  lines  of  the  plane  at  a  d e,  be  extended, 
they  will  meet  in  /,  and  will  extend  the  plane  into  the  tri- 
angle p  d  et,  cutting  the  edge  of  the  butt-joint  in  h,  a  point 
from  which,  if  a  line  be  drawn  upon  the  butt-joint  to  a,  its 
centre,  this  line  will  be  in  the  vertical  plane  pdet,  which 
plane  contains  the  tangent  perpendicular  to  which  the  butt- 
joint  is  drawn ;  consequently  lines  upon  the  butt-joint  par- 
allel to  h  a  will  each  be  in  a  vertical  plane  parallel  to  the 
vertical  tangent  plane,  and  lines  drawn  upon  the  butt-joint 
perpendicular  to  these  lines  will  be  horizontal  lines ;  hence 
the  line  h  a  is  the  required  line  by  which  to  square  the 
wreath  at  the  butt-joint.  Now,  it  will  be  observed  that  the 
triangle  af  a,  is  like  that  given  in  the  various  figures  for  obr 
taining  face-moulds,  to  regulate  the  sliding  of  the  face-mould 
and  the  squaring  at  the  butt-joint.  For  example,  in  Fig. 
163,  the  right-angled  triangle  bul  £v  £vi  is  the  one  referred  to. 
This  triangle  is  in  a  vertical  plane  parallel  to  one  containing 
the  slide-line;  its  longer  side  is  a  vertical  line ;  one  of  the 
sides  containing  the  right  angle  is  equal  to  half  the  thickness 


300 


STAIRS. 


of  the  plank,  while  the  other,  drawn  parallel  to  the  face  of 
the  plank,  is  the  distance  the  face-mould  is  required  to  slide. 
Precisely  like  this,  the  triangle  a  f  at  of  Fig.  165  is  in  the 
vertical  plane  /  b  cn  containing  fg,  the  slide-line  ;  its  longer 
side,  a,  a,  is  a  vertical  line  ;  fa,  one  of  the  sides  containing 
the  right  angle,  is  equal  to  half  the  thickness  of  the  plank, 
while  the  other  side,  drawn  coincident  with  the  surface  of 


FIG.  166, 

the  plank,  is  the  distance  to  slide  the  face-mould.  Therefore 
the  triangle  atf  a  of  Fig.  165  gives  the  required  lines  by 
which  to  regulate  the  application  of  the  face-moulds.  The 
relative  position  of  the  more  important  of  these  lines  is  geo- 
metrically shown  in  Fig.  166,  in  which  A  and  B  are  upon  the 
horizontal  plane  of  the  paper,  C  is  in  a  vertical  plane  stand- 
ing on  the  ground-line  b  d,  and  D  is  a  plan  of  the  butt-joint, 
revolved  upon  the  line  itt  jn  into  the  horizontal  plane,  and 


BLOCKING-OUT   OF   THE   RAIL.  3<DI 

then  perpendicularly  removed  to  the  distance  //,.  The  let- 
tering corresponds  with  that  in  Fig.  165.  The  shaded  part 
of  D  shows  the  end  of  the  squared  wreath.  When  the 
blocked  piece  has  been  marked  by  the  face-mould  in  its 
second  application,  its  edges  are  to  be  trimmed  vertically  as 
shown  in  Fig.  167,  after  which  the  top  and  bottom  surfaces 
of  the  wreath  are  to  be  formed  from  the  shape  marked  on 
the  butt-joints. 


FIG.  167. 

285.— Face-Mould  Curves  are  Elliptical.— The  curves  of 
the  face-mould  for  the  hand-railing  of  any  stairs  of  circular 
plan  are  elliptical,  and  may  be  drawn  by  a  trammel,  or 
in  anv  other  convenient  manner.  The  trouble,  however, 
attending  the  process  of  obtaining  the  axes,  so  as  to  be  able 
to  employ  the  trammel  in  describing  the  curves,  is,  in  many 
cases,  greater  than  it  would  be  to  obtain  the  curves  through 
points  found  by  ordinates,  in  the  usual  manner.  But  as 


302 


STAIRS. 


this  method  for  certain  reasons  may  be  preferred  by  some, 
an  example  is  here  given  in  which  the  curves  are  drawn  by 
a  trammel,  and  in  which  the  method  of  obtaining  the  axes  is 
shown. 

Let  Fig.  168  represent  the  plan  of  a  hand-rail  around  part 


FIG.  168. 

of  a  cylinder  and  with  the  heights  set  up,  the  intersection 
line  obtained,  the  measuring  base-line  drawn,  the  rake-line 
established,  and  the  tangents  on  the  face-mould  located — all 
in  the  usual  manner  as  hereinbefore  described.  Then,  to 
prepare  for  the  trammel,  from  o,  the  centre  of  the  cylinder, 
draw  o  bt  parallel  with  the  intersecting  line,  and  bt  ot  perpen- 


FACE-MOULDS   FOR   ROUND    RAILS.  303 

dicular  to  •£,/,,  the  rake-line  ;  make  btot  equal  to  bo,  and  olal 
equal  to  oa;  through  ot  draw  oth  parallel  with  btfr  From 
o  draw  oe  perpendicular  toob,;  continue  the  central  circular 
line  of  the  rail  around  to  e;  parallel  with  obt  draw  e  fy 
and  from  /„  the  point  of  intersection  of  ef  with  btft,  and 
perpendicular  to  #,/,,  draw  ftet\  make  flel  equal  to  fe\ 
then  0,  is  the  centre  of  the  ellipse,  and  ot  a,  the  semi-conju- 
gate diameter  and  ot  e,  the  semi-transverse  diameter  of  an 
ellipse  drawn  through  the  centre  of  the  face-mould.  To  get 
the  diameters  for  the  edges  of  the  face-mould,  make  a/  c,  and 
a/di  each  equal  to  half  the  width  of  the  rail,  as  at  cad\  par- 
allel to  a  line  drawn  from  at  to  et,  and  through  cr  draw  the 
line  ctg\  also,  parallel  with  a  line  drawn  from  at  to  et  draw 
dth  (see  Art.  559);  then  for  the  curve  at  the  inner  edge  of 
the  face-mould,  otg  is  the  semi-transverse  diameter,  and  olcl 
the  semi-conjugate ;  while  for  the  curve  at  the  outer  edge 
o.h  is  the  semi-transverse  diameter,  and  otdt  the  semi-conju- 
gate. So  much  of  the  edges  of  the  face-mould  as  are  straight 
are  parallel  with  the  tangent.  Now,  placing  the  trammel  at 
the  centre,  as  shown  in  the  figure,  and  making  the  distance 
on  the  rod  from  the  pencil  to  the  first  pin  equal  to  the 
semi-conjugate  diameter,  and  the  distance  to  the  second  pin 
equal  to  the  semi-transverse  diameter,  each  curve  may  be 
drawn  as  shown.  (See  Art.  549.) 

286.— Face-Moulds  for  Round  Hails. — The  previous  ex- 
amples given  for  finding  face-moulds  are  intended  for  moulded 
rails.  For  round  rails  the  same  process  is  to  be  followed, 
with  this  difference :  that  instead  of  finding  curves  on  the 
face-mould  for  the  sides  of  the  rail,  find  one  for  a  centre-line 
and  describe  circles  upon  it,  as  at  Fig.  145 ;  then  trace  the 
sides  of  the  mould  by  the'  points  so  found.  The  thickness  of 
stuff  for  the  twists  of  a  round  rail  is  the  same  as  for  the 
straight  part.  The  twists  are  to  be  sawed  square  through. 

287. — Position  of  the  Butt -Joint. — When  a  block  for 
the  wreath  of  a  hand-rail  is  sawed  square  through  the 
plank,  the  joint,  in  all  cases,  is  to  be  laid  on  the  face-mould 
square  to  the  tangent  and  cut  square  through  the  plank. 


304 


STAIRS. 


Managed  in  this  way,  the  butt-joint  is  in  a  plane  pierced 
perpendicularly  by  the  tangent.  But  if  the  block  be  not 
sawed  square  through,  but  vertically  from  the  edges  of  the 


FIG.  169. 

face-mould,  then,  especially,  care  is  required  in  locating  the 
joint.  The  method  of  sawing  square  through  is  attended 
with  so  many  advantages  that  it  is  now  generally  followed  ; 
yet,  as  it  is  possible  that  for  certain  reasons  some  may  prefer, 


rv 

[(UNI  VET?  si 

POSITION  OF  THE  BUTT-JOINT."  305 

in  some  cases,  to  saw  vertically,  it  is  proper  that  the  method 
of  finding  the  position  of  the  joint  for  that  purpose  should 
be  given.  Therefore,  let  A,  Fig.  169,  be  the  plan  of  the  rail, 
and  B  the  elevation,  showing  its  side;  in  which  kz  is  the 
direction  of  the  butt-joint.  From  k  draw  kb  parallel  to  /<?, 
and  ke  at  right  angles  to  kb\  from  b  draw  b  f,  tending  to 
the  centre  of  the  plan,  and  from  /  draw  fe  parallel  to  bk; 
from  /,  through  e,  draw  I  i,  and  from  i  draw  id  parallel  to 
ef-,  join  dfand  b,  and  db  will  be  the  proper  direction  for  the 
joint  on  the  plan.  The  direction  of  the  joint  on  the  other 
side,  a  c,  can  be  found  by  transferring  the  distances  x  b  and 
od  to  xa  and  oc.  Then  the  allowance  for  over  wood  to 
cover  the  butt-joint  is  shown  as  that  which  is  included  be- 
tween the  lines  ox  and  db.  The  face-mould  must  be  so 
drawn  as  to  cover  the  plan  to  the  line  b  d  for  the  wreath  at 
the  left,  and  to  the  line  a  c  for  that  at  the  right.  By  some 
the  direction  of  the  joint  is  made  to  radiate  toward  the 
centre  of  the  cylinder ;  indeed,  even  Mr.  Nicholson,  in  his 
Carpenter  s  Guide,  so  advised.  That  this  is  an  error  may  be 
shown  as  follows :  In  Fig.  170,  arji  is  the  plan  of  a  part  of  the 
rail  about  the  joint,  s  u  is  the  stretch-out  of  a  i,  and  gp  is  the 
helinet,  or  vertical  projection  of  the  plan  arji.  This  is 
found  by  drawing  a  horizontal  line  from  the  height  set  upon 
each  perpendicular  standing  upon  the  stretch-out  line  su. 
The  lines  upon  the  plan  arji  are  drawn  radiating  to  the 
centre  of  the  cylinder,  and  therefore  correspond  to  the 
horizontal  lines  of  the  helinet  drawn  upon  its  upper  and 
under  surfaces. 

Bisect  rt  on  the  ordinate  drawn  from  the  centre  of  the 
plan,  and  through  the  middle  draw  cb  at  right  angles  to  gv  ; 
from  b  and  c  draw  cd  and  be  at  right  angles  to  su  ;  from  d 
and  e  draw  lines  radiating  toward  the  centre  of  the  plan  ; 
then  do  and  em  will  be  the  direction  of  the  joint  on  the 
plan,  according  to  Nicholson,  and  cb  its  direction  on  the 
falling-mould.  It  must  be  admitted  that  all  the  lines  on  the 
upper  or  the  lower  side  of  the  rail  which  radiate  toward 
the  centre  of  the  cylinder,  as  do,  cm,  or  ij\  are  level;  for 
instance,  the  level  line  wv,  on  the  top  of  the  rail  in  the 
helinet,  is  a  true  representation  of  the  radiating  line//  on 


306 


STAIRS. 


the  plan.  The  line  bh,  therefore,  on  the  top  of  the  rail  in 
the  helinet,  is  a  true  representation  of  e  m  on  the  plan,  and 
kc  on  the  bottom  of  the  rail  truly  represents  do.  From  k 
draw  £/ parallel  to  cb,  and  from  h  draw  hf  parallel  to  bc\ 


FIG.  170. 

join  /  and  b,  also  c  and  /;  then  cklb  will  be  a  true  repre- 
sentation of  the  end  of  the  lower  piece,  B,  and  cfh  b  of  the 
end  of  the  upper  piece,  A  ;  and  fk  or  hi  will  show  how 
much  the  joint  is  open  on  the  inner,  or  concave,  side  of  the 
rail. 


CORRECT   LINES   FOR  BUTT-JOINT. 


307 


To  show  that  the  process  followed  in  Art.  287  is  correct, 
let  do  and  em  (Fig.  171)  be  the  direction  of  the  butt-joint 
found  as  at  Fig.  169.  Now,  to  project,  on  the  top  of  the  rail 
in  the  helinet,  a  line  that  does  not  radiate  toward  the  centre 


of  the  cylinder,  as  jk,  draw  vertical  lines  from  j  and  k  to  w 
and  A,  and  join  w  and  h ;  then  it  will  be  evident  that  wh  is 
a  true  representation  in  the  helinet  of  jk  on  the  plan,  it 
being  in  the  same  plane  as  j  k,  and  also  in  the  same  winding- 
surface  as  wv.  The  line  /»,  also,  is  a  true  representation  on 


308 


STAIRS. 


the  bottom  of  the  helinet  of  the  line  jk  in  the  plan.  The 
line  of  the  joint  e  m,  therefore,  is  projected  in  the  same  way, 
and  truly,  by  ib  on  the  top  of  the  helmet,  and  the  line  do 
by  ca  on  the  bottom.  Join  a  and  i,  and  then  it  will  be  seen 
that  the  lines  c a,  a  t,  and  ib  exactly  coincide  with  cb,  the  line 
of  the  joint  on  the  convex  side  of  the  rail ;  thus  proving  the 
lower  end  of  the  upper  piece,  A,  and  the  upper  end  of  the 
lower  piece,  B,  to  be  in  one  and  the  same  plane,  and  that  the 
direction  of  the  joint  on  the  plan  is  the  true  one.  By  refer- 
ence to  Fig.  169  it  will  be  seen  that  the  line  li  corresponds 
to  xi  in  Fig.  171  ;  and  that  e  k  in  that  figure  is  a  representa- 
tion of  fb,  and  ik  of  db. 

288. — Scrolls  for  Hand-Rail§  :  General  Rule  for  Size 
and  Position  of  the  Regulating  Square.  —  The  breadth 
which  the  scroll  is  to  occupy,  the  number  of  its  revolutions, 
and  the  relative  size  of  the  regulating  square  to  the  eye  of 
the  scroll  being  given,  multiply  the  number  of  revolutions 
by  4,  and  to  the  product  add  the  number  of  times  a  side  of 
the  square  is  contained  in  the  diameter  of  the  eye,  and  the 
sum  will  be  the  number  of  equal  parts  into  which  the  breadth 
is  to  be  divided.  Make  a  side  of  the  regulating  square 
equal  to  one  of  these  parts.  To  the  breadth  of  the  scroll 
add  one  of  the  parts  thus  found,  and  half  the  sum  will  be  the 
length  of  the  longest  ordinate. 


FIG.  172. 

289. — Centres  in  Regulating  Square. — Let  a  2  I  b  (Fig. 
T72)  be  the  size  of  a  regulating  square,  found  according  to 
the  previous  rule,  the  required  number  of  revolutions  being 


SCROLLS   AT  NEWEL. 


309 


if.  Divide  two  adjacent  sides,  as  a  2  and  2  i,  into  as  many 
equal  parts  as  there  are  quarters  in  the  number  of  revolu- 
tions, as  seven  ;  from  those  points  of  division  draw  lines 
across  the  square  at  right  angles  to  the  lines  divided ;  then 
i  being  the  first  centre,  2,  3,4,  5,  6,  and  7  are  the  centres  for 
the  other  quarters,  and  8  is  the  centre  for  the  eye  ;  the  heavy 
lines  that  determine  these  centres  being  each  one  part  less 
in  length  than  its  preceding  line. 


FIG.  173- 

290.— Scroll  for  Hand-Rail  Over  Curtail  Step.  —  Let 
a  b  (Fig.  173)  be  the  given  breadth,  if  the  given  number  of 
revolutions,  and  let  the  relative  size  of  the  regulating  square 
to  the  eye  be  \  of  the  diameter  of  the  eye.  Then,  by  the 
rule,  if  multipled  by  4  gives  7,  and  3,  the  number  of  times  a 
side  of  the  square  is  contained  in  the  eye,  being  added,  the 
sum  is  10.  Divide  a  b,  therefore,  into  10  equal  parts,  and  set 
one  from  b  to  c ;  bisect  a  c  in  e ;  then  a  e  will  be  the  length 
of  the  longest  ordinate  (i  d  or  i  e).  From  a  draw  a  d, 
from  e  draw  e  i,  and  from  b  draw  b  f,  all  at  right  angles  to 
a  b ;  make  e  \  equal  to  a,  and  through  i  draw  i  d  parallel 


310  STAIRS. 

to  a  b ;  set  b  c  from  i  to  2,  and  upon  i  2  complete  the  regu 
lating  square;  divide  this  square  as  at  Fig.  172;  then  de- 
scribe the  arcs  that  compose  the  scroll,  as  follows :  upon  i 
describe  d  e,  upon  2  describe  e  /,  upon  3  describe  /  g, 
upon  4  describe  £-//,  etc.;  make  dl  equal  to  the  width  of  the 
rail,  and  upon  i  describe  / ;//,  upon  2  describe  mn,  etc.;  de- 
scribe the  eye  upon  8,  and  the  scroll  is  completed. 

291. — Scroll  for  Curtail  Step. — Bisect  d  I  (Fig.  173)  in  <?, 
and  make  o  v  equal  to  £  of  the  diameter  of  a  baluster  ;  make 
v  w  equal  to  the  projection  of  the  nosing,  and  e  x  equal  to 
wl\  upon  i  describe  ivy,  and  upon  2  describe  y  z  ;  also,  upon 
2  describe  x  i,  upon  3  describe  ij\  and  so  around  to  z ;  and 
the  scroll  for  the  step  will  be  completed. 

292. — Position  of  Baluster*  Under  Scroll. — Bisect  dl 
(Fig.  173)  in  0,  and  upon  i,  with  i  o  for  radius,  describe  the 
circle  o  r  u  ;  set  the  baluster  at  p  fair  with  the  face  of  the 
second  riser,  r2,  and  from/,  with  half  the  tread  in  the  divi- 
ders, space  off  as  at  o,  q,  r,  s,  /,  u,  etc.,  as  far  as  f ;  upon  2,  3,4, 
and  5  describe  the  centre-line  of  the  rail  around  to  the  eye 
of  the  scroll ;  from  the  points  of  division  in  the  circle  o  r  u 
draw  lines  to  the  centre-line  of  the  rail,  tending  to  8,  the 
centre  of  the  eye ;  then  the  intersection  of  these  radiating 
lines  with  the  centre-line  of  the  rail  will  determine  the  posi- 
tion of  the  balusters,  as  shown  in  the  figure. 

293. — Fal ling-Mould  for  Raking  Part  of  Scroll. — Tangi- 
cal  to  the  rail  at  h  (Fig.  173)  draw  h  k  parallel  to  da ;  then 
ha*  will  be  the  joint  between  the  twist  and  the  other  part 
of  the  scroll.  Make  de*  equal  to  the  stretch-out  of  d  e,  and 
upon  d  e*  find  the  position  of  the  point  k,  as  at  k* ;  at  Fig. 
174,  make  e  d  equal  to  ^2  d  in  Fig.  173,  and  d  c  equal  to  d  c* 
in  that  figure ;  from  c  draw  c  a  at  right  angles  to  e  c,  and 
equal  to  one  rise ;  make  c  b  equal  to  one  tread,  and  from  b, 
through  a,  draw  bj\  bisect  a  c  in  /,  and  through  /  draw  m  q 
parallel  to  eh',  m  q  is  the  height  of  the  level  part  of  a 
scroll,  which  should  always  be  about  3^  feet  from  the  floor ; 
ease  off  the  angle  mfj,  according  to  Art.  521,  and  draw 


FACE-MOULD   FOR  THE   SCROLL. 

gw  n  parallel  to' m  x  j,  and.  at  a  distance  equal  to  the  thick- 
ness of  the  rail ;  at  a  convenient  place  for  the  joint,  as  *, 
draw  in  at  right  angles  to  bj\  through  n  draw///  at  right 
angles  to  e  h ;  make  dk  equal  to  d k~  in  Fig.  173,  and  from  k 
draw  ko  at  right  angles  toe  A;  at  Fig.  173,  make  <//;2  equal 
to  d  h  in  ^.  174,  and  draw  //2  £2  at  right  angles  to 


FIG.   174. 


then  k  a*  and  /i2  b*  will  be  the  position  of  the  joints  on  the 
plan,  and,  at  Fig.  174,  op  and  in  their  position  on  the  falling- 
mould  ;  and  po  in  (Fig.  174)  will  be  the  required  falling- 
mould  which  is  to  be  bent  upon  the  vertical  surface  from  h 2 
to  k  (Fig.  173). 


FIG.  175. 

294.—  Face-Mould  for  the  Scroll.— At  Fig.  173,  from  k 
draw  kr*  at  right  angles  to  r2  d-  at  Fig.  172,  make  h  r 
equal  to  h*  r*  in  Fig.  173,  and  from  r  draw  r  s  at  right 
angles  to  rh  ;  from  the  intersection  of  r  s  with  the  level  line 
m  q,  through  /,  draw  s  t ;  at  Fig.  173,  make  //2  b*  equal  to  ^  /• 
in  Fig.  172,  and  join  b*  and  r2;  from  rt2,  and  from  as  many 


3I2 


STAIRS. 


other  points  in  the  arcs,  a  *  I  and  k  d,  as  is  thought  neces- 
sary, draw  ordinates  to  r2  d  at  right  angles  to  the  latter; 
make  r  b  (Fig.  175)  equal  in  its  length  and  in  its  divisions  to 
the  line  r2  d2  in  Fig.  173 ;  from  r,  n,  o,  p,  q,  and  /  draw  the 
lines  r  k,  n  d,  o  a,  p  e,  q  f,  and  /  c  at  right  angles  to  r  b,  and 
equal  to  r2  k,  d*  s-,f*  a",  etc.,  in  Fig.  173  ;  through  the  points 
thus  found  trace  the  curves  k  I  and  a  c,  and  complete  the 
face-mould,  as  shown  in  the  figure.  This  mould  is  to  be  ap- 
plied to  a  square-edged  plank,  with  the  edge,  /  b  parallel  to 
the  edge  of  the  plank.  The  rake-lines  upon  the  edge  of  the 
plank  are  to  be  made  to  correspond  to  the  angle  s  t  h  in 
Fig.  174.  The  thickness  of  stuff  required  for  this  mould  is 
shown  at  Fig.  174,  between  the  lines  s  t  and  u  v — u  v  being 
drawn  parallel  to  s  t. 


f        7' 


295. — Form  of  Newel-Cap  from  a  Section  of  the  Rail. 

— Draw  a  b  (Fig.  176)  through  the  widest  part  of  the  given 
section,  and  parallel  to  c  d\  bisect  a  b  in  e,  and  through  a,  e, 
and  b  draw  h  i,  f  g,  and  k  j  at  right  angles  to  a  b ;  at  a  con- 


BORING   FOR  BALUSTERS. 


313 


venient  place  on  the  line  fg,  as  o,  with  a  radius  equal  to 
half  the  width  of  the  cap,  describe  the  circle  ijg\  make  r  I 
equal  to  e  b  or  e  a  ;  join  /  and  /,  also  /  and  i\  from  the  curve 
/  b  to  the  line  /  j  draw  as  many  ordinates  as  is  thought 
necessary  parallel  to  fg'  from  the  points  at  which  these 
ordinates  meet  the  line  I  j,  and  upon  the  centre,  0,  describe 
arcs  in  continuation  to  meet  op  ;  from  n  t  x,  etc.,  draw  ns,tu, 
etc.,  parallel  tofg;  make  n  s,  t  u,  etc.,  equal  to  ef,  wv,  etc.; 
make  x  y,  etc.,  equal  to  z  d,  etc.;  make  o  2,  o  3,  etc.,  equal  to 
o  n,  o  t,  etc.;  make  2  4  equal  to  n  s,  and  in  this  way  find  the 
length  of  the  lines  crossing  o  m ;  through  the  points  thus 
found  describe  the  section  of  the  newel-cap  as  shown  in 
the  figure. 


FIG.  177. 

296.— Boring  for  Balusters  in  a  Round  Rail  before  it 
is  Rounded. — Make  the  angle  oct  (Fig.  177)  equal  to  the 
angle  o  c  t  at  Fig.  144;  upon  c  describe  a  circle  with  a 
radius  equal  to  half  the  thickness  of  the  rail ;  draw  the  tan- 
gent b  d  parallel  to  /  c,  and  complete  the  rectangle  e  b  d  f, 
having  sides  tangical  to  the  circle ;  from  c  draw  c  a  at  right 
angles  to  o  c;  then,  b  d  being  the  bottom  of  the  rail,  set  a 
gauge  from  b  to  a,  and  run  it  the  whole  length  of  the  stuff ; 
in  boring,  place  the  centre  of  the  bit  in  the  gauge-mark  at  a, 
and  bore  in  the  direction  a  c.  To  do  this  easily,  make  chucks 
as  represented  in  the  figure,  the  bottom  edge,^/*,  being  par- 
allel to  o  c,  and  having  a  place  sawed  out,  as  e  f,  to  receive 
the  rail.  These  being  nailed  to  the  bench,  the  rail  will  be 
held  steadily  in  its  proper  place  for  boring  vertically.  The 
distance  apart  that  the  balusters  require  to  be,  on  the  under- 
side of  the  rail,  is  one  half  the  length  of  the  rake-side  of  the 
pitch-board. 


STAIRS. 
SPLAYED   WORK. 


297. — The  Bevels  in  Splayed  Work. — The  principles 
employed  in  finding  the  lines  in  stairs  are  nearly  allied  to 
those  required  to  find  the  bevels  for  splayed  work — such  as 
hoppers,  bread-trays,  etc.  A  method  by  which  these  may  be 


FIG.  178. 

obtained  will,  therefore,  here  be  shown.  In  Fig.  178,  a  b  c  is 
the  angle  at  which  the  work  is  splayed,  and  b  d,  on  the 
upper  edge  of  the  board,  is  at  right  angles  to  a  b ;  make  the 
angle  f  g.j  equal  to  a  b  c,  and  from  f  draw  f  h  parallel  to 
e  a ;  from  b  draw  b  o  at  right  angles  to  a  b ;  through  o  draw 
ie  parallel  to  c  b,  and  join  e  and  d\  then  the  angle  a  e  d  will 
be  the  proper  bevil  for  the  ends  from  the  inside,  or  k  d  e 
from  the  outside.  If  a  mitre-joint  is  required,  set  f  g,  the 
thickness  of  the  stuff  on  the  level,  from  e  to  ;«,  and  join  m 
and  d-,  then  kdm  will  be  the  proper  bevil  for  a  mitre-joint. 
If  the  upper  edge  of  the  splayed  work  is  to  be  bevelled, 
so  as  to  be  horizontal  when  the  work  is  placed  in  its  proper 
position,  then  f  gj,  the  same  as  a  b  c,  will  be  the  proper 
bevel  for  that  purpose.  Suppose,  therefore,  that  a  piece  in- 
dicated by  the  lines  kg,  g  f,  and  f  h  were  taken  off ;  then  a 
line  drawn  upon  the  bevelled  surface  from  d  at  right  angles 
to  k  d  would  show  the  true  position  of  the  joint,  because  it 
would  be  in  the  direction  of  the  board  for  the  other  side  ; 
but  a  line  so  drawn  would  pass  through  the  point  o,  thus 
proving  the  principle  correct.  So,  if  a  line  were  drawn  upon 
the  bevelled  surface  from  d  at  an  angle  of  45  degrees  to  k  d, 
it  would  pass  through  the  point  n. 


VIEW   IN   THE  ALHAMBRA. 


SECTION   IV.— DOORS  AND  WINDOWS. 

DOORS. 

298. — General  Requirements. — Among  the  architectural 
arrangements  of  an  edifice,  the  door  is  by  no  means  the  least 
in  importance ;  and  if  properly  constructed,  it  is  not  only 
an  article  of  use,  but  also  of  ornament,  adding  materially  to 
the  regularity  and  elegance  of  the  apartments.  The  dimen- 
sions and  style  of  finish  of  a  door  should  be  in  accordance 
with  the  size  and  style  of  the  building,  or  the  apartment 
for  which  it  is  designed.  As  regards  the  utility  of  doors, 
the  principal  door  to  a  public  building  should  be  of  suffi- 
cient width  to  admit  of  a  free  passage  for  a  crowd  of  people  ; 
while  that  of  a  private  apartment  will  be  wide  enough  if  it 
permit  one  person  to  pass  without  being  incommoded.  Ex- 
perience has  determined  that  the  least  width  allowable  for 
this  is  2  feet  8  inches ;  although  doors  leading  to  inferior 
and  unimportant  rooms  may,  if  circumstances  require  it,  be 
as  narrow  as  2  feet  6  inches ;  and  doors  for  closets,  where  an 
entrance  is  seldom  required,  may  be  but  2  feet  wide.  The 
width  of  the  principal  door  to  a  public  building  may  be 
from  6  to  12  feet,  according  to  the  size  of  the  building;  and 
the  width  of  doors  for  a  dwelling  may  be  from  2  feet  8 
inches  to  3  feet  6  inches.  If  the  importance  of  an  apart- 
ment in  a  dwelling  be  such  as  to  require  a  door  of  greater 
width  than  3  feet  6  inches,  the  opening  should  be  closed 
with  two  doors,  or  a  door  in  two  folds ;  generally,  in  such 
cases,  where  the  opening  is  from  5  to  8  feet,  folding  or  slid- 
ing doors  are  adopted.  As  to*he  height  of  a  door,  it  should 
in  no  case  be  less  than  about  6  feet  3  inches  ;  and  generally 
not  less  than  6  feet  8  inches. 

299. — The  Proportion  between  Width  and  Height:  of 

single  doors,  for  a  dwelling,  should  be  as  2  is  to  5  ;  and,  for 
entrance-doors  to  public  buildings,  as  I  is  to  2.  If  the 
width  is  given  and  the  height  required  of  a  door  for  a 


3i6 


DOORS  AND   WINDOWS. 


dwelling,  multiply  the  width  by  5,  and  divide  the  product 
by  2  ;  but  if  the  height  is  given  and  the  width  required, 
divide  by  5  and  multiply  by  2.  Where  two  or  more  doors 
of  different  widths  show  in  the  same  room,  it  is  well  to  pro- 
portion the  dimensions  of  the  more  important  by  the  above 
rule,  and  make  the  narrower  doors  of  the  same  height  as 
the  wider  ones;  as  all  the  doors  in  a  suit  of  apartments, 
except  the  folding  or  sliding  doors,  have  the  best  appear- 
ance when  of  one  height.  The  proportions  for  folding  or 
sliding  doors  should  be  such  that  the  width  may  be  equal 
to  |-  of  the  height ;  yet  this  rule  needs  some  qualification  ; 
for  if  the  width  of  the  opening  be  greater  than  one  half  the 
width  of  the  room,  there  will  not  be  a  sufficient  space  left 


FIG.  179. 

for  opening  the  doors ;  also,  the  height  should  be  about  one 
tenth  greater  than  that  of  the  adjacent  single  doors. 

300. — Panefls. — Where  doors  have  but  two  panels  in 
width,  let  the  stiles  and  muntins  be  each  \-  of  the  width  ;  or, 
whatever  number  of  panels  there  may  be,  let  the  united 
widths  of  the  stiles  and  the  muntins,  or  the  whole  width  of 
the  solid,  be  equal  to  -|  of  tt^e  width  of  the  door.  Thus :  in 
a  door  35  inches  wide,  containing  two  panels  in  width,  the 
stiles  should  be  5  inches  wide ;  and  in  a  door  3  feet  6  inches 
wide,  the  stiles  should  be  6  inches.  If  a  door  3  feet  6 
inches  wide  is  to  have  3  panels  in  width,  the  stiles  and 
muntins  should  be  each  4^-  inches  wide,  each  panel  being  8 
inches.  The  bottom  rail  and  the  lock-rail  ought  to  be  each 
equal  in  width  to  TV  of  the  height  of  the  door ;  and  the  top 


TRIMMINGS   FOR  DOORS.  317 

rail,  and  all  others,  of  the  same  width  as  the  stiles.  The 
moulding  on  the  panel  should  be  equal  in  width  to  j  of  the 
width  of  the  stile. 

301. — TTrimmings. — Fig.  179  shows  a  method  of  trimming 
doors  :  a  is  the  door-stud ;  b,  the  lath  and  plaster ;  <:,  the 
ground ;  d,  the  jamb  ;  ^,  the  stop ;  /and  g,  architrave  casings  ; 
and  h,  the  door -stile.  It  is  customary  in  ordinary  work  to 
form  the  stop  for  the  door  by  rebating  the  jamb.  But  when 
the  door  is  thick  and  heavy,  a  better  plan  is  to  nail  on  a 
piece  as  at  e  in  the  figure.  This  piece  can  be  fitted  to  the 
door  and  put  on  after  the  door  is  hung  ;  so,  should  the  door 
be  a  trifle  winding,  this  will  correct  the  evil,  and  the  door  be 
made  to  shut  solid. 

Fig.  1 80  is  an  elevation  of  a  door  and  trimmings  suitable 
for  the  best  rooms  of  a  dwelling.  (For  trimmings  generally, 
see  Sect.  V.)  The  number  of  panels  into  which  a  door 
should  be  divided  may  be  fixed  at  pleasure ;  yet  the  present 
style  of  finishing  requires  that  the  number  be  as  small  as  a 
proper  regard  for  strength  will  admit.  In  some  of  our  best 
dwellings,  doors  have  been  made  having  only  two  upright 
panels.  A  few  years'  experience,  however,  has  proved  that 
the  omission  of  the  lock-rail  is  at  the  expense  of  the  strength 
and  durability  of  the  door ;  a  four-panel  door,  therefore,  is 
the  best  that  can  be  made. 

302. — Hanging  l>oor*. — Doors  should  all  be  hung  so  as 
to  open  into  the  principal  rooms ;  and,  in  general,  no  door 
should  be  hung  to  open  into  the  hall,  or  passage.  As  to  the 
proper  edge  of  the  door  on  which  to  affix  the  hinges,  no 
general  rule  can  be  assigned. 


WINDOWS. 

303. — Requirement*  for  Light. — A  window  should  be 
of  such  dimensions,  and  in  such  a  position,  as  to  admit  a 
sufficiency  of  light  to  that  part  of  the  apartment  for  which 
it  is  designed.  No  definite  rule  for  the  size  can  well  be 
given  that  will  answer  in  all  cases ;  yet,  as  an  apprpxima- 


DOORS   AND   WINDOWS. 


tion,  the  following  has  been  used  for  general  purposes. 
Multiply  together  the  length  and  the  breadth  in  feet  of  the 
apartment  to  be  lighted,  and  the  product  by  the  height  in 


FIG.  1 80. 

feet;  then  the  square  root  of   this  product  will  show  the 
required  number  of  square  feet  of  glass. 

304. — Winclow-Frame§. — For  the  size  of  window-frames, 
add  4-J  inches  to  the  width  of  the  glass  for  their  width,  and 


WIDTH   OF  .INSIDE   SHUTTERS.  319 

6£  inches  to  the  height  of  the  glass  for  their  height.  These 
give  the  dimensions,  in  the  clear,  of  ordinary  frames  for  12- 
light  windows ;  the  height  being  taken  at  the  inside  edge  of 
the  sill.  In  a  brick  wall,  the  width  of  the  opening  is  8 
inches  more  than  the  width  of  the  glass — 4^  for  the  stiles  of 
the  sash,  and  3^  for  hanging  stiles — and  the  height  between 
the  stone  sill  and  lintel  is  about  io|  inches  more  than  the 
height  of  the  glass,  it  being  varied  according  to  the  thick- 
ness of  the  sill  of  the  frame. 

305. —  Inside  Shuiter§.  —  Inside  shutters  folding  into 
boxes  require  to  have  the  box-shutter  about  one  inch  wider 
than  the  flap,  in  order  that  the  flap  may  not  interfere  when 
both  are  folded  into  the  box.  The  usual  margin  shown  be- 
tween the  face  of  the  shutter  when  folded  into  the  box  and 
the  quirk  of  the  stop-bead,  or  edge  of  the  casing,  is  half  an 
inch  ;  and,  in  the  usual  method  of  letting  the  whole  of  the 
thickness  of  the  butt  hinge  into  the  edge  of  the  box-shutter, 
it  is  necessary  to  make  allowance  for  the  tlirow  of  the  hinge. 
This  may,  in  general,  be  estimated  at  \  of  an  inch  at  each 
hinging ;  which  being  added  to  the  margin,  the  entire  width 
of  the  shutters  will  be  i  J  inches  more  than  the  width  of  the 
frame  in  the  clear.  Then,  to  ascertain  the  width  of  the  box- 
shutter,  add  i-J  inches  to  the  width  of  the  frame  in  the  clear, 
between  the  pulley-stiles ;  divide  this  product  by  4,  and  add 
half  an  inch  to  the  quotient,  and  the  last  product  will  be 
the  required  width.  For  example,  suppose  the  window  to 
have  3  lights  in  width,  1 1  inches  each.  Then,  3  times  1 1  is 
33,  and  4^  added  for  the  wood  of  the  sash  gives  37^ ;  37^ 
and  1^  is  39,  and  39  divided  by  4  gives  9! ;  to  which  add 
half  an  inch,  and  the  result  will  be  loj  inches,  the  width 
required  for  the  box-shutter. 

306. — Proportion:  Width  and  Height. — In  disposing 
and  locating  windows  in  the  walls  of  a  building,  the  rules  of 
architectural  taste  require  that  they  be  of  different  heights 
in  different  stories,  but  generally  of  the  same  width.  The 
windows  of  the  upper  stories  should  all  range  perpendicu- 
larly over  those  of  the  first,  or  principal,  story ;  and  they 


320  DOORS   AND   WINDOWS. 

should  be  disposed  so  as  to  exhibit  a  balance  of  parts 
throughout  the  front  of  the  building.  To  aid  in  this  it  is 
always  proper  to  place  the  front  door  in  the  middle  of  the 
front  of  the  building  ;  and,  where  the  size  of  the  house  will 
admit  of  it,  this  plan  should  be  adopted.  (See  the  latter 
part  of  Art.  50.)  The  proportion  that  the  height  should 
bear  to  the  width  may  be,  in  accordance  with  general  usage, 
as  follows  : 


of  the  width. 


The  height 

of  basement  windows,  i  -J 

«         i< 

principal-story   " 

2* 

«         n 

second  -story       " 

'1 

«               u 

third-story 

If 

«           « 

fourth-story        " 

I| 

«           « 

attic-story           " 

th( 

the  same  as  the  width. 

But,  in  determining  the  height  of  the  windows  for  the 
several  stories,  it  is  necessary  to  take  into  consideration  the 
.height  of  the  story  in  which  the  window  is  to  be  placed. 
For,  in  addition  to  the  height  from  the  floor,  which  is  gen- 
erally required  to  be  from  28  to  30  inches,  room  is  wanted 
above  the  head  of  the  window  for  the  window-trimming 
and  the  cornice  of  the  room,  besides  some  respectable  space 
which  there  ought  to  be  between  these. 

307. — Circular  Heads. — Doors  and  windows  usually  ter- 
minate in  a  horizontal  line  at  top.  These  require  no  special 
directions  for  their  trimmings.  But  circular-headed  doors 
and  windows  are  more  difficult  of  execution,  and  require 
some  attention.  If  the  jambs  of  a  door  or  window  be  placed 
at  right  angles  to  the  face  of  the  wall,  the  edges  of  the  soffit, 
or  surface  of  the  head,  would  be  straight,  and  its  length  be 
found  by  getting  the  stretch-out  of  the  circle  (Art.  524) ; 
but  when  the  jambs  are  placed  obliquely  to  the  face  of  the 
Avail,  occasioned  by  the  demand  for  light  in  an  oblique 
direction,  the  form  of  the  soffit  will  be  obtained  by  the  fol- 
lowing article ;  as  also  when  the  face  of  the  wall  is  circular, 
as  shown  in  the  succeeding  figure. 


OBLIQUE   SOFFITS   OF  WINDOWS. 


321 


308. — Form   of   Soffit  for  Circular  Window-Heads. — 

When  the  light  is  received  in  an  oblique  direction,  let  abed 
(Fig.  181)  be  the  ground-plan  of  a  given  window,  and  efa  a 
vertical  section  taken  at  right  angles  to  the  face  of  the  jambs. 


FIG.  181. 


From  a,  through  e,  draw  ag  at  righ't  angles  to  ab\  obtain 
the  stretch-out  of  efa,  and  make  eg  equal  to  it;  divide  eg- 
and  efa  each  into  a  like  number  of  equal  parts,  and  drop 
perpendiculars  from  the  points  of  division  in  each ;  from 
the  points  of  intersection,  i,  2,  3,  etc.,  in  the  line  ad, 


FIG.  182. 

draw  horizontal  lines  to  meet  corresponding  perpendicu- 
lars from  eg\  then  those  points  of  intersection  will  give  the 
curve  line  dg,  which  will  be  the  on*3  required  for  the  edge 
of  the  soffit.  The  other  edge,  cht  is  found  in  the  same 
manner. 


322  DOORS  AND   WINDOWS. 

For  the  form  of  the  soffit  for  circular  window-heads, 
when  the  face  of  the  wall  is  curved,  let  abed  (Fig.  182)  be 
the  ground-plan  of  a  given  window,  and  e  f  a  a  vertical  sec- 
tion of  the  head  taken  at  right  angles  to  the  face  of  the 
jambs.  Proceed  as  in  the  foregoing  article  to  obtain  the 
line  dg\  then  that  will  be  the  curve  required  for  the  edge  of 
the  soffit,  the  other  edge  being  found  in  the  same  manner. 

If  the  given  vertical  section  be  taken  in  a  line  with  the 
face  of  the  wall,  instead  of  at  right  angles  to  the  face  of  the 
jambs,  place  it  upon  the  line  cb  (Fig.  181),  and,  having  drawn 
ordinates  at  right  angles  to  cb,  transfer  them  to  efa  ;  in  this 
way  a  section  at  right  angles  to  the  jambs  can  be  obtained. 


SECTION  V.— MOULDINGS  AND  CORNICES. 

MOULDINGS. 

3O9, — mouldings:  are  so  called  because  they  are  of  the 
same  determinate  shape  throughout  their  length,  as  though 
the  whole  had  been  cast  in  the  same  mould  or  form.  The 
regular  mouldings,  as  found  in  remains  of  classic  architec- 
ture, are  eight  in  number,  and  are  known  by  the  following 
names : 


FIG  183.        Annulet,  band,  cincture,  fillet,  listel  or  square. 


FIG.  184.  ^       Astragal  or  bead. 


Torus  or  tore. 


FIG.  185. 


FIG   186  Scotia,  trochilus  or  mouth. 


Ovolo,  quarter-round  or  echinus. 

FIG.  187. 


Cavetto,  cove  or  hollow. 


FIG.   188. 


324  MOULDINGS   AND    CORNICES. 


Cymatium,  or  cyma-recta. 


FIG.  189. 

Ogee. 


Inverted  cymatium,  or  cyma-reversa. 
FIG.  190. 

Some  of  the  terms  are  derived  thus :  Fillet,  from  the  French 
word  fil,  thread.  Astragal,  from  astragalos,  a  bone  of  the 
heel — or  the  curvature  of  the  heel.  Bead,  because  this 
moulding,  when  properly  carved,  resembles  a  string  of  beads. 
Torus,  or  tore,  the  Greek  for  rope,  which  it  resembles  when 
on  the  base  of  a  column.  Scotia,  from  skotia,  darkness,  be- 
cause of  the  strong  shadow  which  its  depth  produces,  and 
which  is  increased  by  the  projection  of  the  torus  above  it. 
Ovolo,  from  ovum,  an  egg,  which  this  member  resembles, 
when  carved,  as  in  the  Ionic  capital.  Cavetto,  from  cavus, 
hollow.  Cymatium,  from  kumaton,  a  wave. 

310. — Characteristics  of  Mouldings. — Neither  of  these 
mouldings  is  peculiar  to  any  one  of  the  orders  of  architect- 
ure ;  and  although  each  has  its  appropriate  use,  yet  it  is  by 
no  means  confined  to  any  certain  position  in  an  assemblage 
of  mouldings.  The  use  of  the  fillet  is  to  bind  the  parts,  as 
also  that  of-  the  astragal  and  torus,  which  resemble  ropes. 
The  ovolo  and  cyma-reversa  are  strong  at  their  upper  ex- 
tremities, and  are  therefore  used  to  support  projecting  parts 
above  them.  The  cyma-recta  and  cavetto,  being  weak  at 
their  upper  extremities,  are  not  used  as  supporters,  but  are 
placed  uppermost  to  cover  and  shelter  the  other  parts.  The 
scotia  is  introduced  in  the  base  of  a  column  to  separate  the 
upper  and  lower  torus,  and  to  produce  a  pleasing  variety 
and  relief.  The  form  of  the  bead  and  that  of  the  torus  is  the 
same ;  the  reasons  for  giving  distinct  names  to  them  are 
that  the  torus,  in  every  order,  is  always  considerably  larger 
than  the  bead,  and  is  placed  among  the  base  mouldings, 


GRECIAN   MOULDINGS. 


325 


whereas  the  bead  is  never  placed  there,  but  on  the  capital  or 
entablature ;  the  torus,  also,  is  seldom  carved,  whereas  the 
bead  is  ;  and  while  the  torus  among  the  Greeks  is  frequently 
elliptical  in  its  form,  the  bead  retains  its  circular  shape.  While 
the  scotia  is  the  reverse  of  the  torus,  the  cavetto  is  the  re- 
verse of  the  ovolo,  and  the  cyma-recta  and  cyma-reversa  are 
combinations  of  the  ovolo  and  cavetto. 


FIG.  191. 

The  curves  of  mouldings,  in  Roman  architecture,  were 
most  generally  composed  of  parts  of  circles ;  while  those  of 
the  Greeks  were  almost  always  elliptical,  or  of  some  one  of 
the  conic  sections,  but  rarely  circular,  except  in  the  case  of 
the  bead,  which  was  always,  among  both  Greeks  and  Ro- 
mans, of  the  form  of  a  semicircle.  Sections  of  the  cone  af- 
ford a  greater  variety  of  forms  than  those  of  the  sphere ;  and 
perhaps  this  is  one  reason  why  the  Grecian  architecture  so 


326 


MOULDINGS  AND   CORNICES. 


much  excels  the  Roman.  The  quick  turnings  of  the  ovolo 
and  cyma-reversa,  in  particular,  when  exposed  to  a  bright 
sun,  cause  those  narrow,  well-defined  streaks  of  light  which 
give  life  and  splendor  to  the  whole. 

311. — A  Profile:  is  an  assemblage  of  essential  parts  and 
mouldings.     That  profile  produces  the  happiest  effect  which 


FIG,  192. 


is  composed  of  but  few  members,  varied  in  form  and  size, 
and  arranged  so  that  the  plane  and  the  curved  surfaces  suc- 
ceed each  other  alternately. 

312. — The  Grecian  Torus  and  Scotia. — Join  the  extremi- 
ties a  and  b  (Fig.  191),  and  from  /,  the  given  projection  of 
the  moulding,  draw/0  at  right  angles  to  the  fillets  ;  from  b 


FJG.  194. 


FIG.  195. 


draw  bh  at  right  angles  to  a  b ;  bisect  a  b  in  c  ;  join  /  and  c, 
and  upon  c,  with  the  radius  cf,  describe  the  arc/^,  cutting 
bh'mh',  through  c  draw  de  parallel  with  the  fillets;  make 
dc  and  ce  each  equal  to  b  //;  then  de  and  a  b  will  be  conju- 
gate diameters  of  the  required  ellipse.  To  describe  the 
curve  by  intersection  of  lines,  proceed  as  directed  at  Art. 


THE   GRECIAN  ECHINUS. 


551  and  note ;  by  a  trammel,  see  Art.  549;  and  to  find  the 
foci,  in  order  to  describe  it  with  a  string,  see  Art.  548. 

313. — The  Grecian  Echinus. — Figs.  192  to  199  exhibit,  va- 
riously modified,  the  Grecian  ovolo,  or  echinus.  Figs.  192  to 
196  are  elliptical,  a  b  and  b  c  being  given  tangents  to  the  curve ; 
parallel  to  which  the  semi-conjugate  diameters,  ad  and  dc, 


IMG.  196. 


FIG.  197. 


are  drawn.  In  Figs.  192  and  193  the  lines  a  d  and  dc  are  semi- 
axes,  the  tangents,  ab  and  be,  being  at  right  angles  to  each 
other.  To  draw  the  curve,  see  Art.  551.  In  Fig.  197  the 
curve  is  parabolical,  and  is  drawn  according  to  Art.  560.  In 
Figs.  198  and  199  the  curve  is  hyperbolical,  being  described 
according  to  Art.  561.  The  length  of  the  transverse  ax's,  a  by 


FIG.  i 


FIG.  199. 


being  taken  at   pleasure  in  order  to  flatten  the  curve,  a  b 
should  be  made  short  in  proportion  to  ac. 

314. — The  Grecian  Cavetto. — In  order  to  describe  this, 
Figs.  200  and  201,  having  the  height  and  projection  given, 
see  Art.  551. 

315. — The  Grecian  Cynia-Rccta. — When  the  projection 
is  more  than  the  height,  as  at  Fig.  202,  make  a  b  equal  to  the 


328 


MOULDINGS  AND   CORNICES. 


height,  and  divide  abed  into  four  equal  parallelograms ;  then 
proceed  as  directed  in  note  to  Art.  551.  When  the  projec- 
tion is  less  than  the  height,  draw  da  (Fig.  203)  at  right  angles 


FIG.  201. 


FIG.  200. 

to  ab\  complete  the  rectangle,  abed;  divide  this  into  four 
equal  rectangles,  and  proceed  according  to  Art.  551. 

316. — The  Grecian  Cyma-Reversa. — When  the  projection 


FIG.  203. 


is  more  than  the  height,  as  at  Fig.  204,  proceed  as  directed 
for  the  last  figure  ;  the  curve  being  the  same  as  that,  the 
position  only  being  changed.  When  the  projection  is  less 


FIG.  204. 


FIG.  205. 


than  the  height,  draw  a  d  (Fig.  205)  at  right  angles  to  the 
fillet ;  make  a  d  equal  to  the  projection  of  the  moulding ;  then 
proceed  as  directed  for  Fig.  202. 


FORMS   OF  ROMAN   MOULDINGS. 


329 


317. — Roman  Mouldings :  are  composed  pf  parts  of  circles, 
and  have,  therefore,  less  beauty  of  form  than  the  Grecian. 
The  bead  and  torus  are  of  the  form  of  the  semicircle,  and  the 
scotia,  also,  in  some  instances ;  but  the  latter  is  often  composed 
of  two  quadrants,  having  different  radii,  as  at  Figs.  206  and 
207,  which  resemble  the  elliptical  curve.  The  ovolo  and  ca- 


FIG.  206. 


FIG.  207. 


vetto  are  generally  a  quadrant,  but  often  less.  When  they  are 
less,  as  at  Fig.  210,  the  centre  is  found  thus :  join  the  extrem- 
ities, a  and  b,  and  bisect  a  b  in  c ;  from  c,  and  at  right  angles 
to  a  b,  draw  c  d,  cutting  a  level  line  drawn  from  a  in  d ;  then  d 
will  be  the  centre.  This  moulding  projects  less  than  its 
height.  When  the  projection  is  more  than  the  height,  as  at 
Fig.  212,  extend  the  line  from  c  until  it  cuts  a  perpendicular 


FIG.  208. 


FIG.  209. 


drawn  from  a,  as  at  d\  and  that  will  be  the  centre  of  the 
curve.  In  a  similar  manner,  the  centres  are  found  for  the 
mouldings  at  Figs.  2QJ,  211,  213,  216,  217,  218,  and  219.  The 
centres  for  the  curves  at  Figs.  220  and  221  are  found  thus: 
bisect  the  line  a  b  at  c  ;  upon  a,  c  and  b  successively,  with  a  c 
or  cb  for  radius,  describe  arcs  intersecting  at  d  and  d\  then 
those  intersections  will  be  the  centres. 


330 


MOULDINGS  AND  CORNICES. 


FIG.  210. 


FIG.  211. 


FIG.  212. 


FIG.  213. 


FIG.  214. 


FIG.  215 


FIG.  216. 


FIG.  217. 


FORMS   OF  MODERN  MOULDINGS. 


331 


3(8. — Modern  Mouldings:  are  represented  in  Figs.  222 
to  229.  They  have  been  quite  extensively  and  successfully 
used  in  inside  finishing.  Fig.  222  is  appropriate  for  a  bed- 
moulding  under  a  low  projecting  shelf,  and  is  frequently 
used  under  mantel-shelves.  The  tangent  i  h  is  found  thus : 
bisect  the  line  ab  at  c,  and  be  at  d-,  from  d  draw  de  at 
right  angles  to  eb\  from  b  draw  bf  parallel  to  ed\  upon  b, 


FIG.  218. 


FIG.  219. 


with  b  d  for  radius,  describe  the  arc  d  f\  divide  this  arc 
into  7  equal  parts,  and  set  one  of  the  parts  from  s,  the  limit 
of  the  projection,  to  o ;  make  o  h  equal  to  o  e ;  from  h,  through 
c,  draw  the  tangent  ki\  divide  b  h,  hc,ci,  and  ia  each  into 
a  like  number  of  equal  parts,  and  draw  the  intersecting  lines 
as  directed  at  Art.  521.  If  a  bolder  form  is  desired,  draw 
the  tangent,  i  h,  nearer  horizontal,  and  describe  an  elliptic 


FIG.  220. 


FIG.  221. 


curve  as  shown  in  Figs.  191  and  224.  Fig.  223  is  much  used 
on  base,  or  skirting,  of  rooms,  and  in  deep  panelling.  The 
curve  is  found  in  the  same  manner  as  that  of  Fig.  222.  In 
this  case,  however,  where  the  moulding  has  so  little  projec- 
tion in  comparison  with  its  height,  the  point  e  being  found 
as  in  the  last  figure,  h  s  may  be  made  equal  to  s  e,  instead  of 
o  e  as  in  the  last  figure.  Fig.  224  is  appropriate  for  a  crown 


332 


MOULDINGS  AND  CORNICES. 


FIG.  223. 


FIG.  224. 


PLAIN   MOULDINGS. 


333 


moulding  of  a  cornice.     In  this  figure  the  height  and  pro- 
jection are  given;  the  direction  of  the  diameter,  ab,  drawn 


FIG.  225. 


FIG.  226. 


through  the  middle  of  the  diagonal,  ef,  is  taken  at  pleasure ; 
and  dc  is  parallel  to  ae.     To  find  the  length  of  dc,  draw  b  h 


FIG.  227. 


FIG.  228. 


FIG.  229. 


at  right  angles  toab;  upon  0,  with  of  for  radius,  describe 
the  arc,  ///,  cutting  bh  in  h;   then  make  o  c  and  od  each 


334 


MOULDINGS   AND    CORNICES. 


equal  to  bh.*  To  draw  the  curve,  see  note  to  Art.  551.  Figs. 
22$  to  229  are  peculiarly  distinct  from  ancient  mouldings, 
being  composed  principally  of  straight  lines  ;  the  few  curves 
they  possess  are  quite  short  and  quick. 

Figs.  230  and  231  are  designs  for  antae  caps.  The  di- 
ameter of  the  antas  is  divided  into  20  equal  parts,  and  the 
height  and  projection  of  the  members  are  regulated  in  ac- 
cordance with  those  parts,  as  denoted  under  H  and  P,  height 
and  projection.  The  projection  is  measured  from  the  mid- 
dle of  the  antse.  These  will  be  found  appropriate  for  por- 
ticos, doorways,  mantelpieces,  door  and  window  trimmings, 


H.P. 


n. 


15 


8l'l4f! 


»*"*i 


910J 


10 


Hi  15 


. 


^|-H 
HH~ 


910J 


FIG.  230. 


FIG.  231. 


etc.  The  height  of  the  antae  for  mantelpieces  should  be 
from  5  to  6  diameters,  having  an  entablature  of  from  2  to 
2J-  diameters.  This  is  a  good  proportion,  it  being  similar  to 
the  Doric  order.  But  for  a  portico  these  proportions  are 


*  The  manner  of  ascertaining  the  length  of  the  conjugate  diameter,  dc,  in 
this  figure,  and  also  in  Figs.  191,  241,  and  242  is' new,  and  is  important  in  this 
application.  It  is  founded  upon  well-known  mathematical  principles,  viz.:  All 
the  parallelograms  that  may  be  circumscribed  about  an  ellipsis  are  equal  to 
one  another,  and  consequently  any  one  is  equal  to  the  rectangle  of  the  two 
axes.  And  again  :  The  sum  of  the  squares  of  every  pair  of  conjugate  diame- 
ters is  equal  to  the  sum  of  the  squares  of  the  two  axes. 


EAVE   CORNICES. 


335 


much  too  heavy  :  ah  antse  1 5  diameters  high  and  an  entab- 
lature of  3  diameters  will  have  a  better  appearance. 

CORNICES. 

319. — Idesigns  for  Cornice§. — Figs.  232  to  240  are  designs 
for  eave  cornices,  and  Figs.  241  and  242  are  for  stucco  cor- 
nices for  the  inside  finish  of  rooms.  In  some  of  these  the 
projection  of  the  uppermost  member  from  the  facia  is 
divided  into  twenty  equal  parts,  and  the  various  members 


FIG.  232. 

are  proportioned  according  to  those  parts,  as  figured  under 
//and  P. 

320.— Eave  Cornices  Proportioned  to  Height  of  Build- 

ing.— Draw  the  line  ac  (Fig.  243),  and  make  be  and  ba  each 
equal  to  36  inches ;  from  b  draw  bdvk  right  angles  to  ac, 
and  equal  in  length  to  f  of  a  c ;  bisect  b  d  in  ^,  and  from  a, 
through  £>,  draw  af\  upon  a,  with  ac  for  radius,  describe  the 
arc  cf,  and  upon  e,  with  ef-iwc  radius,  describe  the  arc/W; 
divide  the  curve  dfc,  into  7  equal  parts,  as  at  IO,  20,  30, 
etc.,  and  from  these  points  of  division  draw  lines  to  be 


336 


MOULDINGS   AND   CORNICES. 


J J 


FIG.  233. 


DDOIfi 


JLJLILJLILIULILJLILJU 


JUUUUULI 


FIG.  234. 


EXAMPLES   OF   CORNICES. 


337 


FIG.  235. 


FIG.  236. 


338 


MOULDINGS  AND   CORNICES. 


FIG.  237. 


H.  P. 


10J 


FIG.  238. 


VARIOUS  DESIGNS  OF  CORNICES. 


339 


H.  P. 


17 


716 


3*31 


FIG.  239. 


H.P. 


FIG.  240. 


340 


MOULDINGS  AND  CORNICES 


H.  P. 


FIG.  241. 


H.  P. 


FIG.  242. 


PROPORTION   OF   CORNICES. 


341 


parallel  to  db;  then  the  distance  b  i  is  the  projection  of  a 
cornice  for  a  building  10  feet  high  ;  b  2,  the  projection  at  20 
feet  high ;  b  3,  the  projection  at  30  feet,  etc.  If  the  projec- 
tion of  a  cornice  for  a  building  34  feet  high  is  required, 
divide  the  arc  between  30  and  40  into  10  equal  parts,  and 


V 


a  b          i        2     3  4     c 

FIG.  243. 

from  the  fourth  point  from  30  draw  a  line  to  the  base,  b  c, 
parallel  with  bd\  then  the  distance  o/  the  point  at  which 
that  line  cuts  the  base  from  b  will  be  the  projection  re- 
quired. So  proceed  for  a  cornice  of  any  height  within  70 
feet.  The  above  is  based  on  the  supposition  that  36  inches 


FIG.  244. 

is  the  proper  projection  for  a  cornice  70  feet  high.  This, 
for  general  purposes,  will  be  found  correct ;  still,  the  length 
of  the  line  be  may  be  varied  to  suit  the  judgment  of  those 
who  think  differently. 

Having  obtained    the   projection  of  a   cornice,  divide  it 
into    20   equal    parts,   and   apportion   the   several   members 


342 


MOULDINGS  AND   CORNICES. 


according  to  its  destination — as  is  shown  at  Figs.  238,  239, 
and  240. 

32 L — Cornice    Proportioned  to  a  given  Cornice. — Let 

the  cornice  at  Fig.  244  be  the  given  cornice.  Upon  any 
point  in  the  lowest  line  of  the  lowest  member,  as  at  #,  with  the 
height  of  the  required  cornice  for  radius,  describe  an  intersect- 
ing arc  across  the  uppermost  line,  as  at  b ;  join  a  and  b ; 
then  b  \  will  be  the  perpendicular  height  of  the  upper  fillet 
for  the  proposed  cornice,  I  2  the  height  of  the  crown  mould- 
ing— and  so  of  all  the  members  requiring  to  be  enlarged  to 
the  sizes  indicated  on  this  line.  For  the  projection  of  the 


\ 


FIG.  245. 

proposed  cornice,  draw  a  d  at  right  angles  to  a  b,  and  c  d  at 
right  angles  to  b  c ;  parallel  with  c  d  draw  lines  from  each 
projection  of  the  given  cornice  to  the  line  ad-,  then  e  d  \v\\\ 
be  the  required  projection  for  the  proposed  cornice,  and  the 
perpendicular  lines  falling  upon  c  d  will  indicate  the  proper 
projection  for  the  members. 

To  proportion  a  cornice  according  to  a  larger  given  cor- 
nice, let  A  (Fig.  245)  be  the  given  cornice.  Extend  a  o  to  b, 
and  draw  c  d  at  right  angles  to  a  b ;  extend  the  horizontal 
lines  of  the  cornice,  A,  until  they  touch  o  d\  place  the  height 
of  the  proposed  cornice  from  o  to  e,  and  join /"and  e\  upon 
o,  with  the  projection  of  the  given  cornice,  o  a,  for  radius, 


TO   FIND   THE   ANGLE   BRACKET. 


describe  the  quadrant  ad\  from  d draw  db  parallel  tofe; 
upon  ot  with  o  b  for  radius,  describe  the  quadrant  b  c ;  then 
o  c  will  be  the  proper  projection  for  the  proposed  cornice. 
Join  a  and  c ;  draw  lines  from  the  projection  of  the  different 
members  of  the  given  cornice  to  ao  parallel  to  od\  from 
these  divisions  on  the  line  ao  draw  lines  to  the  line  oc 
parallel  to  a  c ;  from  the  divisions  on  the  line  of  draw  lines 
to  the  line  o e  parallel  to  the  line  fe\  then  the  divisions  on 
the  lines  o  e  and  o  c  will  indicate  the  proper  height  and  pro- 
jection for  the  different  members  of  the  proposed  cornice. 
In  this  process,  we  have  assumed  the  height,  o  e,  of  the  pro- 
posed cornice  to  be  given;  but  if  the  projection,  o  c,  alone 


FIG.  247. 


be  given,  we  can  obtain  the  same  result  by  a  different  pro- 
cess. Thus:  upon  o,  with  oc  for  radius,  describe  the  quad- 
rant c  b ;  upon  <?,  with  o  a  for  radius,  describe  the  quadrant 
ad\  join  d  and  b ;  from  /draw  fe  parallel  to.db\  then  oe 
will  be  the  proper  height  for  the  proposed  cornice,  and  the 
height  and  projection  of  the  different  members  can  be 
obtained  by  the  above  directions.  By  this  problem,  a  cor- 
nice can  be  proportioned  according  to  a  smaller  given  one 
as  well  as  to  a  larger  ;  but  the  method  described  in  the  pre- 
vious article  is  much  more  simple  for  that  purpose. 

322.— Angle   Bracket  in  a  Built  Cornice.— Let  A  (Fig. 
246)  be  the  wall  of  the  building,  and  B  the  given  bracket, 


344 


OULDINGS  AND   CORNICES. 


which,  for  the  present  purpose,  is  turned  down  horizontally. 
The  angle-bracket,  C,  is  obtained  thus :  through  the  ex- 
tremity, #,  and  parallel  with  the  wall,  fd,  draw  the  line  a  b  ; 
make^r  equal  af,  and  through  c  draw  cb  parallel  with  ed\ 
join  d  and  £,  and  from  the  several  angular  points  in  B  draw 
ordinates  to  cut  db  in  i,  2,  and  3 ;  at  those  points  erect  lines 
perpendicular  to  db',  from  h  draw  kg  parallel  to  fa-,  take 


FIG.    248. 

the  ordinates,  i  0,  2  0,  etc.,  at  B,  and  transfer  them  to  C,  and 
the  angle-bracket,  C,  will  be  denned.  In  the  same  manner, 
the  angle-bracket  for  an  internal  cornice,  or  the  angle-rib  of 
a  coved  ceiling,  or  of  groins,  as  at  Fig.  247,  can  be  found. 

323.— Raking  mouldings  matched  with  Level  Returns— 

Let  A  (Fig.  248)  be  the  given  moulding,  and  A  b  the  rake  of 


CROWN   MOULDING  ON  THE   RAKE.  345 

the  roof.  Divide  the  curve  of  the  given  moulding  into  any 
number  of  parts,  equal  or  unequal,  as  at  i,  2,  and  3  ;  from 
these  points  draw  horizontal  lines  to  a  perpendicular  erected 
from  c ;  at  any  convenient  place  on  the  rake,  as  at  B,  draw 
a  c  at  right  angles  to  A  b  ;  also  from  b  draw  the  horizontal 
line  b  a ;  place  the  thickness,  da,  of  the  moulding  at  A  from 
b  to  a,  and  from  a  draw  the  perpendicular  line  a  e ;  from 
the  points  i,  2,  3,  at  A,  draw  lines  to  C  parallel  to  A  b ; 
make  a  i,  a  2,  and  a  3,  at  B,  and  at  C,  equal  to  ^ui,  etc.,  at  A  ; 
through  the  points,  i,  2,  and  3,  at  B,  trace  the  curve — this 
will  be  the  proper  form  for  the  raking  moulding,  From  i, 
2,  and  3,  at  C,  drop  perpendiculars  to  the  corresponding 
ordinates  from  i,  2,  and  3,  at  A  ;  through  the  points  of  inter- 
section, trace  the  curve — this  will  be  the  proper  form  for  the 
return  at  the  top. 


' 

PART    II. 


SECTION  VI.— GEOMETRY. 


324. — Mathematics  Essential. — In  this  and  the  following 
Sections,  which  will  constitute  Part  II.,  there  are  treated  of 
certain  matters  which  may  be  considered  as  elementary. 
They  are  all  very  necessary  to  be  understood  and  acquired 
by  the  builder,  and  are  here  compactly  presented  in  a  shape 
which,  it  is  believed,  will  aid  him  in  his  studies,  and  at  the 
same  .time  prove  to  be  a  great  convenience  as  a  matter  of 
reference. 

The  many  geometrical  forms  which  enter  into  the 
composition  of  a  building  suggest  a  knowledge  of  .Elemen- 
tary Geometry  as  essential  to  an  intelligent  comprehension 
of  its  plan  and  purpose.  One  of  the  prime  requisites  of  a 
building  is  stability,  a  quality  which  depends  upon  a  proper 
distribution  of  the  material  of  which  the  building  is  con- 
structed ;  hence  a  knowledge  of  the  laws  of  pressure  and 
the  strength  of  materials  is  essential ;  and  as  these  are  based 
upon  the  laws  of  proportion  and  are  expressed  more  con- 
cisely in  algebraic  language,  a  knowledge  of  Proportion  and 
of  Algebra  are  likewise  indispensable  to  a  comprehensive 
understanding  of  the  subject.  There  will  be  found  in  this 
work,  however,  only  so  much  of  these  parts  of  mathematics 
as  have  been  deemed  of  the  most  obvious  utility  in  the 
Science  of  Building.  For  a  more  exhaustive  treatment  of 
the  subjects  named,  the  reader  is  referred  to  the  many  able 
works,  readily  accessible,  which  make  these  subjects  their 
specialties. 

325.— Elementary  Geometry.— In  all  reasoning  defini- 
tions are  necessary,  in  order  to  insure  in  the  minds  of  the 


348  GEOMETRY. 

proponent  and  respondent  identity  of  ideas.  A  corollary  is 
an  inference  deduced  from  a  previous  course  of  reasoning. 
An  axiom  is  a  proposition  evident  at  first  sight.  In  the  fol- 
lowing demonstrations  there  are  many  axioms  taken  for 
granted  (such  as,  things  equal  to  the  same  thing  are  equal  to 
one  another,  etc.) ;  these  it  was  thought  not  necessary  to 
introduce  in  form. 

326. — Definition. — If  a  straight  line,  as  A  B  (Fig.  249), 
stand  upon  another  straight  line,  as  CD,  so  that  the  two 


angles  made  at  the  point  B  are  equal — A  B  C  to  A  B D  (Art. 
499,  obtuse  angle] — then  each  of  the  two  angles  is  called  a 
right  angle. 

327. — Definition.— The  circumference  of  every  circle 
is  supposed  to  be  divided  into  360  equal  parts,  called 
degrees ;  hence  a  semicircle  contains  180  degrees,  a  quad- 
rant 90,  etc. 


C  B  D 

FIG.  250. 

328. — Definition. — The  measure  of  an  angle  is  the  num- 
ber of  degrees  contained  between  its  two  sides,  using  the 
angular  point  as  a  centre  upon  which  to  describe  the  arc. 
Thus  the  arc  C  E  (Fig.  250)  is  the  measure  of  the  angle 
C  B  E,  E  A  of  the  angle  E  B  A,  and  A  D  of  the  angle  A  B  D. 

329. — Corollary. — As  the  two  angles  at  B  (Fig.  249)  are 
right  angles,  and  as  the  semicircle,  CAD,  contains  180  de- 
grees (Art.  327),  the  measure  of  two  right  angles,  therefore,  is 


HE 

'UNIVERSITY 

RI6HT  ANGLES   AND    OBLIQUE   ANG^§^ 

^4,  .._. 

1 80  degrees ;  of  one  right  angle,  90  degrees  ;  of  half  a  right 
angle,  4$  ;  of  one  third  of  a  right  angle,  30,  etc. 

330. — Definition. — In  measuring  an  angle  (Art.  328),  no 
regard  is  to  be  had  to  the  length  of  its  sides,  but  only  to  the 
degree  of  their  inclination.  Hence  equal  angles  are  such  as 
have  the  same  degree  of  inclination,  without  regard  to  the 
length  of  their  sides. 

331. — Axiom.— If  two  straight  lines  parallel  to  one 
another,  as  A  B  and  CD  (Fig.  251),  stand  upon  another 
straight  line,  as  E  F,  the  angles  A  B  F  and  CDF  are  equal, 
and  the  angle  A  B  E  is  equal  to  the  angle  CD  E. 

332. — Definition. — If  a  straight  line,  as  A  B  (Fig.  250), 
stand  obliquely  upon  another  straight  line,  as  CD,  then  one 

A  C 

/ 


B          ~  D 

FIG.  251. 


of  the  angles,  as  A  B  C,  is  called  an  obtuse  angle,  and  the 
other,  as  A  B  D,  an  acute  angle. 

333.— Axiom.— The  two  angles  ABDzndABC  (Fig.  250) 
are  together  equal  to  two  right  angles  (Arts.  326,  329) ;  also, 
the  three  angles  A  B  D,  E  B  A,  and  CBE  are  together 
equal  to  two  right  angles. 

334.. corollary. — Hence   all    the   angles   that   can   be 

made  upon  one  side  of  a  line,  meeting  in  a  point  in  that 
line,  are  together  equal  to  two  right  angles. 

335, corollary. — Hence  all  the  angles  that  can  be  made 

on  both  sides  of  a  line,  at  a  point  in  that  line,  or  all  the 
angles  that  can  be  made  about  a  point,  are  together  equal  to 
four  right  angles. 


350 


GEOMETRY. 


336. — Proposition. — If  to  each  of  two  equal  angles  a 
third  angle  be  added,  their  sums  will  be  equal.  Let  ABC 
and  D  E  F  (Fig.  252)  be  equal  angles,  and  the  angle  I J  K  the 
one  to  be  added.  Make  the  angles  G  B  A  and  H  E  D  each 
equal  to  the  given  angle  IJ  K\  then  the  angle  G  B  C  will  be 
equal  to  the  angle  HE  F;  for  if  ABC  and  D  E  F  be  angles 


A~ 


C 

FIG.  252. 


of  90  degrees,  and  IJK  30,  then  the  angles  GBC  and 
HEF  will  be  each  equal  to  90  and  30  added,  viz.,  120 
degrees. 

337. — Proportion. — Triangles  that  have  two  of  their 
sides  and  the  angle  contained  between  them  respectively 
equal,  have  also  their  third  sides  and  the  two  remaining 


FIG.  253. 

angles  equal ;  and  consequently  one  triangle  will  every  way 
equal  the  other.  Let  ABC  (Fig.  253)  and  DEF  be  two 
given  triangles,  having  the  angle  at  A  equal  to  the  angle  at 
D,  the  side  A  B  equal  to  the  side  D  E,  and  the  side  A  C 
equal  to  the  side  D  F\  then  the  third  side  of  one,  B  C,  is  equal 
to  the  third  side  of  the  other,  E  F\  the  angle  at  B  is  equal  to 
the  angle  at  £*  and  the  angle  at  C  is  equal  to  the  angle  at 


EQUAL  TRIANGLES   IN   PARALLELOGRAMS.  351 

F.  For  if  one  triangle  be  applied  to  the  other,  the  three 
points  B,  A,  C,  coinciding  with  the  three  points  E,  D,  F,  the 
line  Bf  must  coincide  with  the  line  EF\  the  angle  at  B 
with  the  angle  at  E;  the  angle  at  C  with  the  angle  at  F\ 
and  the  triangle  B  A  C  be  every  way  equal  to  the  triangle 
EDF. 

338. — Proposition. — The  two  angles  at  the  base  of  an 
isosceles  triangle  are   equal.      Let  ABC  (Fig.  254)  be  an 


B  D  C 

FIG.  254. 

isosceles  triangle,  of  which  the  sides,  A  B  and  A  C,  are  equal. 
Bisect  the  angle  (Art.  506)  BA  C  by  the  line  A  D.  Then,  the 
\ineJ5A  being  equal  to  the  line  A  C,  the  line  A  D  of  the 
triangle  E  being  equal  to  the  line  A  D  of  the  triangle  F 
(being  common  to  each),  the  angle  BAD  being  equal  to  the 
angle  DA  C, — the  line  B  D  must,  according  to  Art.  337,  be 


D      D  c 

FIG.  255. 

equal  to  the  line  D  C,  and  the  angle  at  B  must  be  equal  to 
the  angle  at  C. 

339. — Proportion. — A  diagonal  crossing  a  parallelogram 
divides  it  into  two  equal  triangles.  Let  C  D  E  F  (Fig.  255) 
be  a  given  parallelogram,  and  C  F  a  line  crossing  it  diag- 
onally. Then,  as  E  C  is  equal  to  F  D,  and  EF  to  CD,  the 
angle  at  E  to  the  angle  at  D,  the  triangle  A  must,  according 
to  Art.  337,  be  equal  to  the  triangle  B. 


352 


GEOMETRY. 


340 — Proposition.— Let  J KL  M  (Fig.  256)  be  a  given 
parallelogram,  and  K L  a  diagonal.  At  any  distance  between 
J K and  L  M  draw  N P  parallel  to  J K\  through  the  point 
G,  the  intersection  of  the  lines  KL  and  N  P,  draw  HI 
parallel  to  K  M.  In  every  parallelogram  thus  divided,  the 
parallelogram  A  is  equal  to  the  parallelogram  B.  For,  ac- 
cording to  Art.  339,  the  triangle  JKL  is  equal  to  the  tri- 
angle K  L  M,  the  triangle  C  to  the  triangle  D,  and  E  to  F\ 


M 


this  being  the  case,  take  D  and  F  from  the  triangle  K  LM, 
and  C  and  £  from  the  triangle  JKL,  and  what  remains  in 
one  must  be  equal  to  what  remains  in  the  other ;  therefore, 
the  parallelogram  A  is  equal  to  the  parallelogram  B. 

34-1. — Proposition. — Parallelograms  standing  upon  the 
same  base  and  between  the  same  parallels  are  equal.  Let. 
A  BCD  and  EFCD  (Fig.  257)  be  given  parallelograms 


FIG.  257. 

standing  upon  the  same  base,  CD,  and  between  the  same 
parallels,  A  F  and  CD.  Then  A  B  and  E  F,  being  equal  to 
CD,  are  equal  to  one  another;  BE  being  added  to  both 
A  B  and  E  F,  A  E  equals  B  F;  the  line  A  C  being  equal  to 
B  D,  and  A  E  to  B  F,  and  the  angle  C  A  E  being  equal  (Art. 
331)  to  the  angle  D  B  F,  the  triangle  A  E  C  must  be  equal 
(Art.  337)  to  the  triangle  B  F  D\  these  two  triangles  being 
equal,  take  the  same  amount,  the  triangle  B  E  G,  from  each, 


TRIANGLE   EQUAL  TO   QUADRANGLE. 


353 


and  what  remains  in  one,  A  B  G  C,  must  be  equal  to  what 
remains  in  the  other,  E  F  D  G\  these  two  quadrangles  being 
equal,  add  the  same  amount,  the  triangle  C  G  D,  to  each,  and 
they  must  still  be  equal ;  therefore,  the  parallelogram 
A  B  CD  is  equal  to  the  parallelogram  E  F  C  D. 

342. — Corollary. — Hence,  if  a  parallelogram  and  triangle 
stand  upon  the  same  base  and  between  the  same  parallels, 


H 


D 

FIG.  258. 


the  parallelogram  will  be  equal  to  double  the  triangle. 
Thus,  the  parallelogram  A  D  (Fig.  257)  is  double  (Art.  339) 
the  triangle  C  E  D. 

343. proposition.— Let  FG  H D  (Fig.  258)  be  a  given 

quadrangle   with   the   diagonal  F  D.     From    G  draw    G  E 


FIG.  259. 

parallel  toFD-  extend  HD  to  E\  join  F  and  E ;  then  the 
triangle  .F^^will  be  equal  in  area  to  the  quadrangle 
FGHD.  For  since  the  triangles  FDG  and  FDE  stand 
upon  the  same  base,  F  D,  and  between  the  same  parallels, 
FD  and  G  E,  they  are  therefore  equal  (Arts.  341,  342);  and 
since  the  triangle  C  is  common  to  both,  the  remaining  tri- 


354  GEOMETRY. 

angles,  A  and  B,  are  therefore  equal ;  then,  B  being  equal  to 
A,  the  triangle  F E  H  is  equal  to  the  quadrangle  F  G  H  D. 

344. — Proposition. — If  two  straight  lines  cut  each  other, 
as  FG  and  H  J  (Fig.  259),  the  vertical,  or  opposite  angles, 
A  and  C,  are  equal.  Thus,  FE,  standing  upon  H  y,  forms 
the  angles  B  and  C,  which  together  amount  (Art.  333)  to  two 
right  angles  ;  in  the  same  manner,  the  angles  A  and  B  form 
two  right  angles ;  since  the  angles  A  and  B  are  equal  to  B 
and  C,  take  the  same  amount,  the  angle  B,  from  each  pair, 
and  what  remains  of  one  pair  is  equal  to  what  remains  of 
the  other ;  therefore,  the  angle  A  is  equal  to  the  angle  C. 
The  same  can  be  proved  of  the  opposite  angles  B  and  D. 


345. — Propo8ition. — The  three  angles  of  any  triangle  are 
equal  to  two  right  angles.  Let  ABC  (Fig.  260)  be  a  given 
triangle,  with  its  sides  extended  to  F,  E  and  D,  and  the  line 
C  G  drawn  parallel  to  BE.  As  G  C  is  parallel,  to  EB,  the 
angle  at //is  equal  (Art.  331)  to  the  angle  at  L  ;  as  the  lines 
FC  and  BE  cut  one  another  at  A,  the  opposite  angles  at  M 
and  TV"  are  equal  (Art.  334) ;  as  the  angle  at  N  is  equal  (Art. 
331)  to  the  angle  at  Jy  the  angle  at  J  is  equal  to  the  angle  at 
M]  therefore,  the  three  angles  meeting  at  £7  are  equal  to  the 
three.angles  of  the  triangle  A  B  C ;  and  since  the  three  angles 
at  C  are  equal  (Art.  333)  to  two  right  angles,  the  three  angles 
of  the  triangle  ABC  must  likewise  be  equal  to  two  right 
angles.  Any  triangle  can  be  subjected  to  the  same  proof. 

346. — Corollary. — Hence,  if  one  angle  of  a  triangle  be  a 
right  angle,  the  other  two  angles  amount  to  just  one  right 
angle. 


RIGHT  ANGLE  IN  SEMICIRCLE.  355 

34-7. — Corollary. — If  one  angle  of  a  triangle  be  a  'right 
angle  and  the  two  remaining  angles  are  equal  to  one  another, 
these  are  each  equal  to  half  a  right  angle. 

348. — Corollary. — If  any  two  angles  of  a  triangle  amount 
to  a  right  angle,  the  remaining  one  is  a  right  angle. 

349. — Corollary. — If  any  two  angles  of  a  triangle  are  to- 
gether equal  to  the  remaining  angle,  that  remaining  angle  is 
a  right  angle. 

350. — Corollary. — If  any  two  angles  of  a  triangle  are  each 
equal  to  two  thirds  of  a  right  angle,  the  remaining  angle  is 
also  equal  to  two  thirds  of  a  right  angle. 

351. — Corollary. — Hence,  the  angles  of  an  equilateral 
triangle  are  each  equal  to  two  thirds  of  a  right  angle. 


FIG.  261. 

352. — Proposition. — If  from  the  extremities  of  the  di- 
ameter of  a  semicircle  two  straight  lines  be  drawn  to  any 
point  in  the  circumference,  the  angle  formed  by  them  at  that 
point  will  be  a  right  angle.  Let  ABC  (Fig.  261)  be  a  given 
semicircle  ;  and  A  B  and  B  C  lines  drawn  from  the  extrem- 
ities of  the  diameter  A  C  to  the  given  point  B;  the  angle 
formed  at  that  point  by  these  lines  is  a  right  angle.  Join 
the  point  B  and  the  centre  D  ;  the  lines  DA,  D B,  and  D  C, 
being  radii  of  the  same  circle,  are  equal;  the  angle  at A  is, 
therefore,  equal  (Art.  338)  to  the  angle  at  E\  also,  the  angle 
at  C  is,  for  the  same  reason,  equal  to  the  angle  at  F\  the 
angle  A  B  C,  being  equal  to  the  angles  at  A  and  C  taken  to- 
gether,  must,  therefore  (Art.  349),  be  a  right  angle. 

353. — Proportion. — The  square  on  the  hypothenuse  of  a 
right-angled  triangle  is  equal  to  the  squares  on  the  two  re- 


356' 


GEOMETRY. 


maining  sides.  Let  ABC  (Fig.  262)  be  a  given  right-angled 
triangle,  having  a  square  formed  on  each  of  its  sides ;  then 
the  square  BE  is  equal  to  the  squares  NCand  GB  taken 
together.  This  can  be  proved  by  showing-  that  the  parallelo- 
gram B  L  is  equal  to  the  square  G B ;  and  that  the  parallelo- 
gram C L  is  equal  to  the  square  H C.  The  angle  C B D  is  a 
right  angle,  and  the  angle  A  B  F\s  a  right  angle  ;  add  to  each 
of  these  the  angle  ABC]  then  the  angle  F  B  C  will  evidently 
be  equal  (Art.  336)  to  the  angle  ABD\  the  triangle  FB  C 
and  the  square  G  B,  being  both  upon  the  same  base,  F  B,  and 
between  the  same  parallels,  F  B  and  G  C,  the  square  G  B  is 
equal  (A  rt.  342)  to  twice  the  triangle  F B  C ';  the  triangle 
A  B  D  and  the  parallelogram  B  L,  being  both  upon  the  same 


FIG.  262. 

base,  B  D,  and  between  the  same  parallels,  B  D  and  A  L,  the 
parallelogram  BL  is  equal  to  twice  the  triangle  A  B  D ; 
the  triangles,  FB  C  and  AB  D,  being  equal  to  one  another 
(Art.  337),  the  square  G  B  is  equal  to  the  parallelogram  B  L, 
either  being  equal  to  twice  the  triangle  F  B  C  or  A  B  D.  The 
method  of  proving  H  C  equal  to  C  L  is  exactly  similar — thus 
proving  the  square  B  E  equal  to  the  squares  H  C  and  G  B, 
taken  together. 

This  problem,  which  is  the  4/th  of  the  First  Book  of 
Euclid,  is  said  to  have  been  demonstrated  first  by  Pythago- 
ras. It  is  stated  (but  the  story  is  of  doubtful  authority) 
that  as  a  thank-offering  for  its  discovery  he  sacrificed  a  hun- 
dred oxen  to  the  gods.  From  this  circumstance  it  is  some- 
times called  the  Jiecatomb  problem.  It  is  of  great  value  in 


DIAGONAL   OF   SQUARE   FORMING   OCTAGON. 


357 


the  exact  sciences,  more  especially  in  Mensuration  and  As- 
tronomy, in  which  many  otherwise  intricate  calculations  are 
by  it  made  easy  of  solution. 

354. — Proposition. — In  an  equilateral  octagon  the  semi- 
diagonal  of  a  circumscribed  square,  having  its  sides  coinci- 
dent with  four  of  the  sides  of  the  octagon,  equals  the  dis- 
tance along  a  side  of  the  square  from  its  corner  to  the  more 
remote  angle  of  the  octagon  occurring  on  that  side  of  the 
square.  Let  Fig.  263  represent  the  square  referred  to ;  in 
which  0  is  the  centre  of  each  ;  then  A  O  equals  A  D.  To 
prove  this,  it  need  only  be  shown  that  the  triangle  A  O  D  is 
an  isosceles  triangle  having  its  sides  A  O  and  A  D  equal.  The 


FIG.  263. 

octagon  being  equilateral,  it  is  also  equiangular,  therefore 
the  angles  BCO,ECO,ADO,  etc.,  are  all  equal.  Of  the 
right-angled  triangle  FEC,FC  and  FE  being  equal,  the 
two  angles  FECand  FCE,  are  equal  (Art.  338),  and  are 
therefore  (Art.  347)  each  equal  to  half  a  right  angle.  In  like 
manner  it  may  be  shown  that  FA  B  and  FR  A  are  also  each 
equal  to  half  a  right  angle.  And  since  FE  C  and  FA  B  are 
equal  angles,  therefore  the  lines  E  C  and  A  B  are  parallel 
(Art.  331,)  and  hence  the  angles  E  CO  and  A  OD  are  equal. 
These  being  equal,  and  the  angles  ECO  and  A  D  O  being 
equal  by  construction,  as  before  shown,  therefore  the  angles 
A  OD  and  A  D  0  are  equal,  and  consequently  the  lines  A  O 
and  A  D  are  equal.  (Art.  338.) 


35$  GEOMETRY. 

355. — Proposition. — An  angle  at  the  circumference  of  a 
circle  is  measured  by  half  the  arc  that  subtends  it ;  that  is, 
the  angle  A  B  C  (Fig.  264)  is  equal  to  half  the  angle  A  D  C. 
Through  the  centre  D  draw  the  diameter  BE.  The  tri- 
angle A  B  D  is  an  isosceles  triangle,  A  Z?and  B  D  being  ra- 
dii, and  therefore  equal ;  hence,  the  two  angles  at  F  and  G 
are  equal  (Art.  338),  and  the  sum  of  these  two  angles  is  equal 
to  the  angle  at  H  (Art.  345),  and  therefore  one  of  them,  Gt  is 
equal  to  the  half  of  H.  The  angles  at  H  and  at  G  (or  A  BE) 
are  both  subtended  by  the  arc  A  E.  Now,  since  the  angle 


FIG.  264. 

at  H  is  measured  by  the  arc  A  E,  which  subtends  it,  there- 
fore the  half  of  the  angle  at  H  would  be  measured  by  the 
half  of  the  arc  A  E ;  and  since  G  is  equal  to  the  half  of  H, 
therefore  G  or  A  BE  is  measured  by  the  half  of  the  arc  A  E. 
It  maybe  shown  in  like  manner  that  the  angle  E  B  C  is 
measured  by  half  the  arc  E  C,  and  hence  it  follows  that  the 
angle  A  B  C  is  measured  by  half  the  arc,  A  C,  that  sub- 
tends it. 

356. — Proposition. — In  a  circle  all  the  inscribed  angles, 
A,  B,  and  C  (Fig.  265),  which  stand  upon  the  same  side  of  the 


EQUAL  ANGLES   IN   CIRCLES. 


359 


chord  DE  are  equal.     For  each  angle  is  measured  by  half 
the  arc  D F E  (Art.  355).     Hence  the  angles  are  all  equal. 

357. — Corollary. — Equal  chords,  in  the  same  circle,  sub- 
tend equal  angles. 


FIG.  265. 


358. — Proposition. — The  angle  formed  by  a  chord  and 
tangent  is  equal  to  any  inscribed  angle  in  the  opposite  seg- 


ment of  the  circle ;  that  is,  the  angle  D  (Fig.  266)  equals  the 
angle  A.  Let  H F  be  the  chord,  and  E  6  the  tangent ;  draw 
the  diameter  y H ';  then  JH  G  is  a  right  angle,  also  J  F  H  \$ 


3  6o 


GEOMETRY. 


a  right  angle.  (Art.  352.)  The  angles  A  and  B  together  equal 
a  right  angle  (Art.  346) ;  also  the  angles  B  and  D  together 
equal  a  right  angle  (equal  to  the  angle  J HG)  ;  therefore,  the 
sum  of  A  and  B  equals  the  sum  of  B  and  D.  From  each  of 
these  two  equals,  taking  the  like  quantity  B,  the  remainders 
A  and  D  are  equal.  Thus,  it  is  proved  for  the  angle  at  A ; 
it  is  also  true  for  any  other  angle ;  for,  since  all  other  in- 
scribed angles  on  that  side  of  the  chord  line  H  F  equal  the 
angle  A  (Art.  356),  therefore  the  angle  formed  by  a  chord 
and  tangent  equals  any  angle  in  the  opposite  segment  of  the 
circle.  This  being  proved  for  the  acute  angle  D,  it  is  also 
true  for  the  obtuse  angle  EHF-,  for,  from  any  point,  K  (Fig. 
267)  in  the  arcHKF,  draw  lines  to  7,  F  and  H ;  now,  if  it  can 


be  proved  that  the  angle  EH  F  equals  the  angle  FK H,  the 
entire  proposition  is  proved,  for  the  angle  FKH  equals  any 
of  all  the  inscribed  angles  that  can  be  drawn  on  that  side  of 
the  chord.  (Art.  356.)  To  prove,  then,  that  EH  F  equals 
H  KF\  the  angle  EH  F  equals  the  sum  of  the  angles  A  and 
B  ;  also  the  angle  H  K  F  equals  the  sum  of  the  angles  C  and 
D.  The  angles  B  and  D,  being  inscribed  angles  on  the  same 
chord,  J  F,  are  equal.  The  angles  C  and  A,  being  right  angles 
(Art.  352),  are  likewise  equal.  Now,  since  A  equals  C  and  B 
equals  D,  therefore  the  sum  of  A  and  B  equals  the  sum  of  C 
and  D—OY  the  angle  E  H  F  equals  the  angle  H  K  F. 

359. — Propo§kion.  —  The   areas   of   parallelograms  of 
equal  altitude  are  to  each  other  as  the  bases  of  the  parallelo- 


PARALLELOGRAMS   PROPORTIONATE   TO   BASES. 


361 


grams.  In  Fig.  268  the  areas  of  the  rectangles  A  B  CD  and 
B  E  D  F  are  to  each  other  as  the  bases  CD  and  D F.  For, 
putting  the  two  bases  in  form  of  a  fraction  and  reducing  this 
fraction  to  its  lowest  terms,  then  the  numerator  and  denomi- 
nator of  the  reduced  fraction  will  be  the  numbers  of  equal 
parts  into  which  the  two  bases  respectively  may  be  divided. 
For  example,  let  the  two  given  bases  be  12  and  9  feet  respect- 
ively, then  -ijp-  =  f ,  and  this  gives  four  parts  for  the  larger 
base  and  three  parts  for  the  smaller  one.  So,  in  Fig.  268, 
divide  the  base  CD  into  four  equal  parts,  and  the  base  D  F 
into  three  equal  parts ;  then  the  length  of  any  one  of  the 
parts  in  CD  will  equal  the  length  of  any  one  of  the  parts  in 
D  F.  Now,  parallel  with  A  C,  draw  lines  from  each  point  of 
division  to  the  line  A  E.  These  lines  will  evidently  divide 
the  whole  figure  into  seven  equal  parts,  four  of  them  occupy- 


FIG.  268. 

ing  the  area  A  B  C  D,  and  three  of  them  occupying  the  area 
B  E  D  F.  Now  it  is  evident  that  the  areas  of  the  two  rect- 
angles are  in  proportion  as  the  number  of  parts  respectively 
into  which  the  base-lines  are  divided,  or  that — . 

A  B  C  D  \  B  E  D  F  \  :  C  D  \  D  F. 

The  areas  in  this  particular  case  are  as  4  to  3.  But  in  gen- 
eral the  proportion  will  be  as  the  lengths  of  the  bases. 
Thus  the  proposition  is  proved  in  regard  to  rectangles,  but 
it  has  been  shown  (Art.  341)  that  all  parallelograms  of  equal 
base  and  altitude  are  equal.  Therefore  the  proposition  is 
proved  in  regard  to  parallelograms  generally,  including  rect- 
angles. 

360. — Proposition. — Triangles  of  equal  altitude  are  to 
each  other  as  their  bases.     It  has  been  shown  (Art.  359)  that 


362  GEOMETRY. 

parallelograms  of  equal  altitude  are  in  proportion  as  their 
bases,  and  it  has  also  been  shown  (Art.  342)  that  of  a  triangle 
and  parallelogram,  when  of  equal  base  and  altitude,  the 
parallelogram  is  equal  to  double  the  triangle.  Therefore 
triangles  of  equal  altitude  are  to  each  other  as  their  bases. 

361. —  Proposition. — Homologous  triangles  have  their 
corresponding  sides  in  proportion.  Let  the  line  CD  (Fig. 
269)  be  drawn  parallel  with.  A  B.  Then  the  angles  E  CD 
and  E  A  B  are  equal  (Art.  331),  also  the  angles  E  D  C  and 
E  B  A  are  equal.  Therefore  the  triangles  E  CD  and  E  A  B 
are  homologous,  or  have  their  corresponding  angles  equal. 


FIG.  269. 

For,  join  C  to-  B^  and  A  to  A  then  the  triangles  A  C  D  and 
B  C  D)  standing  on  the  same  base,  C  D,  and  between  the 
same  parallels,  CD  and  A  B,  are  equal  in  area.  To  each 
of  these  equals  join  the  common  area  C  D  E,  and  the  sums 
A  DE  and  B  CE  will  be  equal.  The  triangles  CD  E  and 
A  D  E,  having  the  same  altitude,  are  to  each  other  as 
their  bases  C  E  and  A  E  (Art.  360),  or— 

CDE  :  ADE  :  :  CE  :  A  E. 

Also  the  triangles  CDE  and  B  C  E,  having  the  same  alti- 
tude, are  to  each  other  as  their  bases  D  E  and  BE,  or — 

CDE  ;  BCE  :  :  DE  :  BE. 


CHORDS   GIVING   EQUAL   RECTANGLES. 


363 


And,  since  the  triangles  A  D  E  and  B  C E  are  equal,  as  before 
shown,  therefore,  substituting  in  the  last  proportion  A  D  E 
for  B  CE,  we  have — 

CD  E  :  AD  E  :  :  D  E  :  BE. 

The  first  two  factors  here  being  identical  with  the  first  two 
in  the  first  proportion  above,  we  have,  comparing  the  two 
proportions — 

CE  :  AE  :  :  DE  :  B  E\ 

or,  we  have  the  corresponding  sides  of  one  triangle,  CD  E, 
in  proportion  to  the  corresponding  sides  of  the  other,  A  BE. 


FIG.  270. 

362. — Proposition. — Two  chords,  E  F  and  CD  (Fig.  270), 
intersecting,  the  parallelogram  or  rectangle  formed  by  the 
two  parts  of  one  is  equal  to  the  rectangle  formed  by  the 
two  parts  of  the  other.  That  is,  the  product  of  C  G  multi- 
plied by  G  D  is  equal  to  the  product  of  E  G  multiplied  by 
G  F.  The  triangle  A  is  similar  to  the  triangle  B,  because  it 
has  corresponding  angles.  The  angle  H  equals  the  angle  G 
(Art.  344);  the  angle  at  J  equals  the  angle  at  K,  because 
they  stand  upon  the  same  chord,  D  F  (Art.  356) ;  for  the  same 


364  GEOMETRY. 

reason  the  angle  M  equals  the  angle  L,  for  each  stands  upon 
the  same  chord,  E  C.  Therefore,  the  triangle  A  having  the 
same  angles  as  the  triangle  B,  the  length  of  the  sides  of  one 
are  in  like  proportion  as  the  length  of  the  sides  in  the  other 
(Art.  361).  So— 

CG  :  EG  :  :   GF  :   G  D. 

Hence,  as  the  product  of  the  means  equals  the  product  of 
the  extremes  (Art.  373),  E  G  multiplied  by  GF  is  equal  to 
C  G  multiplied  by  G  D. 

363. — Proposition. — If  the  sides  of  a  quadrangle  are 
bisected,  and  lines  drawn  joining  the  points  of  bisection  in 
the  adjacent  sides,  these  lines  will  form  a  parallelogram. 


FIG.   271. 

Draw  the  diagonals  A  B  and  CD  (Fig.  271).  It  will  here  be 
perceived  that  the  two  triangles  A  E  O  and  A  C  D  are  homol- 
ogous, having  like  angles  and  proportionate  sides.  Two  of 
the  sides  of  one  triangle  lie  coincident  with  the  two  corres- 
ponding sides  of  the  other  triangle,  therefore  the  contained 
angles  between  these  sides  in  each  triangle  are  identical. 
By  construction,  these  corresponding  sides  are  proportion- 
ate;  AC  being  equal  to  twice  A  E,  and  A  D  being  equal  to 
twice  A  O ;  therefore  the  remaining  sides  are  proportionate, 
CD  being  equal  to  twice  E  O,  hence  the  remaining  corres- 
ponding angles  are  equal.  Since,  then,  the  angles  A  E  O 
and  A  C  D  are  equal,  therefore  the  line  E  0  is  parallel  with 


PARALLELOGRAM    IN   QUADRANGLE.  365 

the  diagonal  CD — so,  likewise,  the  line  MNis  parallel  to  the 
same  diagonal,  CD.  If,  therefore,  these  two  lines,  EO  and 
MN,  are  parallel  to  the  same  line,  CD,  they  must  be  parallel 
to  each  other.  In  the  same  manner  the  lines  ON  and  EM 
are  proved  parallel  to  the  diagonal  A  B,  and  to  each  other ; 
therefore  the  inscribed  figure  ME  ON  is  a  parallelogram. 
It  may  be  remarked,  also,  that  the  parallelogram  so  formed 
will  contain  just  one  half  the  area  of  the  circumscribing 
quadrangle. 


SECTION  VII.— RATIO,  OR  PROPORTION. 

364. — Merchandise. — A  carpenter  buys  9  pounds  of  nails 
for  45  cents.  He  afterwards  buys  87  pounds  at  the  same 
rate.  How  much  did  he  pay  for  them  ? 

An  answer  to  this  question  is  readily  found  by  multiply- 
ing the  87  pounds  by  45  cents,  the  price  of  the  9  pounds, 
and  dividing  the  product,  3915,  by  9 ;  the  quotient,  435  cents, 
is  the  answer  to  the  question. 

365. — The  "Rule  of  Three." — The  process  by  which 
this  problem  is  solved  is  known  as  the  Rule  of  Three,  or 
Proportion. 

In  cases  of  this  kind  there  are  three  quantities  given,,  to 
find  a  fourth.  Previous  to  working  the  question  it  is  usual 
to  make  a  statement,  placing  the  three  given  quantities  in 
such  order  that  the  quantity  which  is  of  like  kind  with  the 
answer  shall  occupy  the  second  place;  the  quantity  upon 
which  this  depends  for  its  value  is  put  m  the  first  place,  and 
the  remaining  quantity,  which  is  of  like  kind  with  that  in  the. 
first  place,  is  assigned  to  the  third  place. 

When  thus  arranged,  the  second  and  third  quantities  are 
multiplied  together  and  the  product  is  divided  by  the  first 
quantity  ;  the  quotient,  the  answer  to  the  question,  is  a 
fourth  quantity.  These  four  quantities  are  related  to  each 
other  in  this  manner,  namely :  the  first  is  in  proportion  to 
the  second  as  the  third  is  to  the  fourth  ;  or,  taking  the 
quantities  of  the  given  example,  and  putting  them  in  a  for- 
mal statement  with  the  customary  marks  between  them,  we 
have — 

9  :  45   :  :  87  :  435, 

which  is  read:  9  is  to  45  as  87  is  to  435  ;  or,  9  is  in  propor- 
tion to  45  as  87  is  to  435  ;  or,  9  bears  the  same  relation  to  45 
as  87  does  to  435. 


EQUALITY   OF   RATIOS.  367 

366.  —  Couple*:   Antecedent,  Consequent.  —  These    four 
quantities  are  termed  Proportionals,  and  may  be  divided  into 
two  couples  ;    the   first  and  second   quantities  forming  one 
couple,  and  the  third  and  fourth  the  other  couple.    Of  each 
couple  the  first  quantity  is  termed  the  antecedent,  and  the 
last  the  consequent.     Thus  9  is  an  antecedent  and  45  its  con- 
sequent ;  so,  also,  87  is  an  antecedent  and  435  its  consequent. 

367.  —  Equal  Couples  :  an  Equation.  —  These  four  quan- 
tities may  be  put  in  form  thus  : 

45.  =  435 
9   -  87 

Each  couple  is  here  stated  as  a  fraction  :  each  has  its  ante- 
cedent  beneath  its  consequent,  and  the  two  couples  are 
separated  by  a  sign,  two  short  parallel  lines,  signifying 
equality.  This  is  an  equation,  and  is  read  thus  :  45  divided 
by  9  is  equal  to  435  divided  by  87  ;  or,  as  ordinary  fractions  : 
45  ninths  are  equal  to  435  eighty-sevenths. 

368.  —  Equality  of  Ratios.  —  Each  couple  is  also  termed  a 
Ratio,  and  the  two  the  Equality  of  Ratios.     Thus  the  ratio 


—is  equal  to  the  ratio  ^-.      If  the   division  indicated    in 
9  87 

these  two  ratios  be  actually  performed,  the  equality  between 
the  two  will  at  once  be  apparent,  for  the  quotient  in  each 
case  is  5.  The  resolution  of  each  couple  into  its  simplest 
form  by  actual  division  is  shown  thus  : 


f- 

These  are  read  :  45  divided  by  9  equals  5  ;    and  435  divided 
by  87  equals  5. 

369.  —  Equali  multiplied  by  Equal§  Give  Equals.  —  If  two 

equal  quantities  be  each  multiplied  by  a  given  quantity,  the 


368  RATIO,   OR   PROPORTION. 

two  products  will  be  equal.  For  example,  the  fractions  f 
and  f  are  each  equal  to  ^,  and  are  therefore  equal  to  each 
other.  If  these  two  equal  quantities  be  each  multiplied  by 
any  given  number,  say,  for  example,  by  4,  we  shall  have  4 
times  f  equals  f ,  and  4  times  f  equals  -1/- ;  these  products,  f 
and  \2-  are  each  equal  to  2,  and  therefore  equal  to  each 
other. 

370. — Multiplying  an  Equation. — The  quantity  on  each 
side  of  the  sign  =  is  called  a  member  of  the  equation.  If 
each  member  be  multiplied  by  the  same  quantity,  the 
equality  of  the  two  members  is  not  thereby  disturbed  (Art. 
369);  therefore,  if  the  two  members  of  the  equation 

45  —  -415.  (Art.  367)  be  each  multiplied  by  87,  or  be  modified 
9         °7 
thus: 

45x87^435x87 
9      ,          87 

in  which  x,  the  sign  for  multiplication,  indicates  that  the 
quantities  between  which  it  is  placed  are  to  be  multiplied 
together ;  this  ddition  to  each  member  of  the  equation  does 
not  destroy  the  equality  ;  the  members  are  still  equal, 
though  considerably  enlarged.  The  equality  may  be  easily 
tested  by  performing  the  operations  indicated  in  the  equa- 
tion. For  example  :  for  the  first  member,  we  have  45  times 
87  equals  3915,  and  this  divided  by  9  equals  435.  Again,  for 
the  second  member  we  have  435  times  87  equals  37845,  and 
this  divided  by  87  equals  435,  the  same  result  as  that  for  the 
first  member.  Thus  the  multiplication  has  not  interfered 
with  the  equality  of  the  members. 

371. — Multiplying  and  Dividing  one  Member  of  an 
Equation:  Cancelling. — If  a  quantity  be  multiplied  by  a 
given  number,  and  the  product  be  divided  by  the  same 
given  number,  the  quotient  will  equal  the  original  quantity. 
For  example  :  if  8  be  multiplied  by  3,  the  product  will  be  24 ; 
then  if  this  product  be  divided  by  3,  the  quotient  will  be  8, 
the  original  quantity.  Thus  the  value  of  a  quantity  is  not 


TRANSFERRING   FACTORS   IN   EQUATIONS.  369 

changed  by  multiplying  it  by  a  number,  provided  it  be  also 
divided  by  the  same  number. 

From  this,  also,  we  learn  that  the  value  of  a  quantity 
which  is  required  to  be  multiplied  and  divided  by  the  same 
number  will  not  be  changed  if  the  multiplication  and  divis- 
ion be  both  omitted  ;  one  cancels  the  other.  Therefore  the 
number  87,  appearing  in  the  second  member  of  the  equation 
in  the  last  article  both  as  a  multiplier  and  a  divisor,  may  be 
omitted  without  destroying  the  equality  of  the  two  mem- 
bers. The  equation  thus  treated  will  be  reduced  to  — 

45  x87 


This  expression  is  read  :  the  product  of  45  times  87  divided 
by  9  equals  435.  It  will  be  observed  that  we  have  here  the 
four  terms  of  the  problem  in  Art.  365,  three  of  them  in  the 
first  member,  and  the  fourth,  the  answer  to  the  problem,  in 
the  second  member. 

372.  —  Transferring  a  Factor  .—Each  of  the  four  quan- 
tities in  the  aforesaid  equation  is  termed  a  factor.  Compar- 
ing the  equation  of  the  last  article  with  that  of  Art.  43,  it 
appears  that  the  two  are  alike  excepting  that  the  factor  87 
has  been  transferred  from  one  member  of  the  equation  to 
the  other,  and  that,  whereas  it  was  before  a  divisor,  it  has 
now  become  a  multiplier.  From  this  we  learn  that  a  factor 
may  be  transferred  from  one  member  of  an  equation  to  the 
other,  provided  that  in  the  transfer  its  relative  position  to 
the  horizontal  line  above  or  below  it  be  also  changed  ;  that 
is,  if,  before  the  transfer,  it  be  below  the  line,  it  must  be  put 
above  the  line  in  the  other  member;  or,  if  above  the  line,  it 
must  be  put  below,  in  the  other  member.  For  example  :  in 
the  equation  of  the  last  article  let  the  factor  9  be  removed 
to  the  second  member  of  the  equation.  It  stands  as  a  divi- 
sor in  the  first  member  ;  therefore,  by  the  rule,  it  must  appear 
as  a  multiplier  after  the  transfer  ;  or  — 

45  x  87  =  9  x  435; 


3/O  RATIO,   OR   PROPORTION. 

which  is  read,  45  times  87  equals  9  times  435.  By  actually 
performing  the  operations  here  indicated,  we  find  that  each 
member  gives  the  same  product,  3915;  thus  proving  that 
the  equality  of  the  two  members  was  not  interfered  with 
by  the  transfer. 

373. — Equality  of  Products:  Means  and  Extremes.— In 

Art.  366,  the  four  factors  are  put  in  the  usual  form  of  four 
proportionals.  A  comparison  of  these  with  the  four  factors 
as  they  appear  in  the  equation  in  the  last  article,  shows  that 
the  first  member  contains  the  second  and  third  of  the  four 
proportionals,  and  the  second  member  contains  the  first  and 
the  fourth  ;  or,  the  first  contains  what  are  termed  the 
means,  and  the  second,  the  extremes.  From  this  we  learn 
that  in  any  set  of  four  proportionals,  the  product  of  the 
means  equals  the  product  of  the  extremes.  As  for  example, 

-  =  i^ ;  so,  also,  -  =  i£,   an  equality   of    ratios  :    hence  the 

four  factors,  2,  3,  4,  6,  are  four  proportionals,  and  may  be 
put  thus : 

Extreme,  /nean,  mean,  extreme. 
2:31:4:6 

and,  as  above  stated,  the  product  of  the  means  (3x4=)  12, 
equals  the  product  of  the  extremes  (2x6=)  12. 

374-.  —  Homologous     Triangles    Proportionate.  —  The 

discussion  of  the  subject  of  Ratios  has  thus  far  been  con- 
fined to  its  relations  with  the  mercantile  problem  of  Art. 
364.  The  rules  of  proportion  or  the  equality  of  ratios 
apply  equally  to  questions  other  than  those  of  a  mercantile 
character.  They  apply  alike  to  all  questions  in  which  quan- 
tities of  any  kind  are  comparable.  For  example,  in  geome- 
try, lines,  surfaces,  and  solids  bear  a  certain  fixed  relation  to 
one  another,  and  are,  therefore,  fit  subjects  for  the  rules  of 
proportion.  It  is  shown,  in  Art.  361,  that  the  correspond- 
ing sides  of  homologous  triangles  are  in  proportion  to  one 
another.  Hence,  when,  of  two  similar  triangles,  two  corres- 
ponding sides  and  one  other  side  are  given,  then  by  the 
equality  of  ratios  the  side  corresponding  to  this  other  side 


RATIOS   APPLIED   TO   TRIANGLES.  3/1 

may  be  computed.  For  example  :  in  two  triangles,  such  as 
ECD  and  EAB  (Fig.  269),  having  their  corresponding 
angles  equal,  let  the  side  E  C,  in  the  triangle  ECD,  equal  12 
feet,  and  the  corresponding  side  E  A,  in  the  triangle  E  A  B, 
equal  16  feet,  and  the  side  ED,  of  triangle  ECD,  equal  14 
feet.  Now,  having  these  three  sides  given,  how  can  we  find 
the  fourth  ?,  Putting  them  in  proportion,  we  have,  as  in 
Art.  361— 

CE  :  AE  :  :  DE  :  BE', 

and,  substituting  for  the  known  sides,  their  dimensions,  we 
have  — 

12  :   16  :  :   14  :  B  E  ; 

and,  by  Art.  373— 

12  x  BE  =  16  x  14. 

Dividing  each  member  by  12,  gives  — 


Performing   the  multiplication  and  division  indicated,  we 
have  — 


Thus  we  have  the  fourth  side.equal  to  i8|  feet. 

375.  _  The  Steelyard.  —  An  example  of  lour  proportion- 
als may  also  be  found  in  the  relation  existing  between  the 
arms  of  a  lever  and  the  weights  suspended  at  their  ends.  A 
familiar  example  of  a  lever  is  seen  in  the  common  steelyard 
used  by  merchants  in  weighing  goods.  This  is  a  bar,  A  B, 
of  steel,  arranged  as  in  Fig.  272,  with  hooks  and  links,  and  a 
suspended  platform  to  carry  R,  the  article  to  be  weighed  ; 
and  with  a  weight  P,  suspended  by  a  link  at  B,  from  the  bar 
A  B,  along  which  the  weight  P  is  movable. 

The  entire  load  is  sustained  by  links  attached  to  the  ful- 
crum, or  point  of  suspension  C.  The  apparatus  is  in  equi- 
librium without  R  and  P.  In  weighing  any  article,  R,  the 


3/2  RATIO,    OR   PROPORTION. 

weight  P  is  moved  along  the  bar  B  C  until  the  weight  just 
balances  the  load,  or  until  the  bar  A  B  will  remain  in  a  hori- 
zontal position.  If  the  weight  P  be  too  far  from  the  fulcrum 
C  the  end  of  the  bar  B  will  fall,  but  if  it  be  too  near  it  will 
rise. 

376. — The  L<ever  Exemplified   l>y  the   Steelyard.  —  To 

exemplify  the  principle  of  the  lever,  let  the  bar  A  B  (Fig. 
272)  be  balanced  accurately  with  the  scale  platform,  but 
without  the  weights  R  and  P.  Then,  placing  the  article  R 
upon  the  platform,  move  the  weight  P  along  the  beam  until 
there  is  an  equilibrium.  Suppose  the  distances  A  C  and  B  C 
are  found  to  be  2  and  40  inches  respectively,  and  suppose 


FIG.  272. 

the  weight  P  to  equal  5  pounds,  what  at  this  point  will  be 
the  weight  of  R?  By  trial  we  shall  find  that  R  =  100  pounds. 
Again,  if  a  portion  of  R  be  removed,  then  the  weight  P 
would  have  to  be  moved  along  the  bar  B  C  to  produce  an 
equilibrium  ;  suppose  it  be  moved  until  its  distance  from  C 
be  found  to  be  20  inches,  then  the  weight  of  R  would  be 
found  to  be  50  pounds,  or — 

R  =  50  pounds. 

Again,  suppose  a  part  of  the  weight  taken  from  R  be  re- 
stored, and  the  weight  P,  on  being  moved  to  a  point  re- 
quired for  equilibrium,  be  found  to  measure  30  inches  from 
C,  then  we  shall  find  that— 

R  =  75  pounds. 


RATIOS   OF  THE   LEVER.  373 

Thus  when  — 

B  C  =  40,  R  =  loo  ;  or,  --  =  2  •  5  ; 

40 


BC=2o,  R  =  $o>J  or,      -=2-5; 

showing  an  equality  of  ratios  ;  or,  in  general,  B  C  is  in  pro- 
portion to  R)  or  — 

BC  :  R. 

If,  instead  of  moving  P  along  B  C,  its  position  be  permanent, 
and  the  weight  P  be  reduced  as  needed  to  produce  equilib- 
rium with  the  various  articles,  R,  which  in  turn  may  be 
put  upon  the  scale  ;  then  we  shall  find  that  if  when  the 
weight  P  equals  5  pounds  the  article  R  equals  100,  and  there 
is  an  equilibrium,  then  when  — 

P-——  x  5  =4-5,  R  will  equal  -^*x  100  =-90; 

8  8 

P  =  —  x  5  =  4,  R  will  equal  —  x  100  =  80; 

P=  -^  x  5  r=  3  •  5,  R  will  equal  —  x  100  =  70  ; 
and  so  on  for  other  proportions;  and  in  every  case  we  shall 

r> 

have  the  ratio  -    equal  20,  thus  — 


R        90 

-75-  =  '^—  =  20 ' 

P       4-5 

R        80 
P=T  =  2°; 

^?  70 

7  =  3^  =  2°- 


374  RATIO,   OR  PROPORTION. 

Thus   we    have   an   equality  of   ratios    in   comparing    the 
weights. 

Again,  if  the  weight  P  and  the  article  R  be  permanent  in 
weight,  and  the  distances  A  C,  B  C  be  made  to  vary,  then  if 
there  be  an  equilibrium  when  A  C  is  2  and  B  C  is  40,  we 
shall  find  that  when  — 

o  O 

A  C  =  —  -  x  2  =  I  -6  ;  B  C  will  equal  —  x  40  =  32  , 
A  C  '  =  —  x  2  =  i  •  2  ;  /?  C"  will  equal  —  x  40  =  24  ; 

AC=  ~  x2  =  °'8;  BC  wil1  eclual  —  x  40  =  16  ; 

and  so  on  for  other  proportions,  and  in  every  case  we  shall 

BC 

have  the  ratio  -jyr  =  20  ;  thus  — 

B  C        32 

_    -       ^  -    *}f\  . 

-     _,-,   ™~  --•  •     —  •  v/  • 

^4  T       1-6 


^C~  0-8  • 

producing  thus  an  equality  of  ratios  in  comparing  the  arms 
of  the  lever.  From  these  experiments  we  have  found,  in 
comparing  the  article  weighed  with  an  arm  of  the  lever,  the 
constant  ratio  B  C  :  R,  and  when  comparing  the  weights 
we  have  found  the  constant  ratio  P  :  R.  Again,  in  com- 
paring the  arms  of  the  lever,  we  find  the  constant  ratio 
A  C  :  B  C.  Putting  two  of  these  couples  in  proportion,  we 
have  — 

A  C  :  B  C  :  :  P  :  R. 

Hence  (Art.  373)  we  have— 


PRINCIPLE   OF   THE    LEVER   DEMONSTRATED.  375 

Dividing  both  members  by  A  C,  we  have  — 

BC*P 
~AC— 

In  a  steelyard  the  short  arm,  A  C,  and  the  weight,  or  poise, 
P,  are  unvarying  ;  therefore  we  have— 


or,  when  ^  „•  is  constant,  we  have  — 

R  :  B  C. 

377.—  The  L,cvcr  Principle  Bcmon§trated.  —  The  rela- 
tion between  the  weights  and  their  arms  of  leverage  may  be 
demonstrated  as  follows  :  * 


FIG.   273.  . 

Let  A  B  G  H,  Fig.  273,  represent  a  beam  of  homogeneous 
material,  of  equal  sectional  area  throughout,  and  suspended 
upon  an  axle  or  pin  at  C,  its  centre.  This  beam  is  evidently 
in  a  state  of  equilibrium.  Of  the  part  of  the  beam  A  D  G  K, 
let  E  be  the  centre  of  gravity ;  and  of  the  remaining  part, 
D  D  K  H,  let  F  be  the  centre  of  gravity. 

If  the  weight  of  tne  material  in  A  D  G K\>t  concentrated 
at  E,  its  centre  of  gravity,  and  the  weight  of  the  material  in 


*  The  principle  upon  which  this  demonstration  is  based  may  be  found  in  an 
article  written  by  the  author  and  published  in  the  Mathematical  Monthly,  Cam- 
bridge, U.  S.,  for»i8s8,  p.  77. 


376  RATIO,    OR   PROPORTION. 

DBKH  be  concentrated  in  F,  its  centre  of  gravity,  the 
state  of  equilibrium  will  not  be  interfered  with.  Therefore 
let  the  ball  R  be  equal  in  weight  to  the  part  A  D  G  K,  and 
the  ball  P  equal  to  the  weight  of  the  part  D BKH\  and  let 
these  two  balls  be  connected  by  the  rod  E  F.  Then  these 
two  balls  and  rod,  supported  at  C,  will  evidently  be  in  a 
state  of  equilibrium  (the  rod  EF  being  supposed  to  be  with- 
out weight). 

Now,  it  is  proposed  to  show  that  R  is  to  P  as  C F  is  to 
C E.  This  can  be  proved;  for,  since  R  equals  the  area 
ADGK  and  P  equals  the  area  DBKH,  therefore  R  is  in 
proportion  to  A  D,  as  P  is  to  D B  (Art.  359) ;  or,  taking  the 
halves  of  these  lines,  R  is  in  proportion  to  A  J  as  P  is  to 
LB. 

Also,  J  L  equals  half  the  length  of  the  beam  ;  for  J D  is 
the  half  of  A  D,  and  D  L  is  the  half  of  DB;  thus  these  two 
parts  (JD  +  DL)  equal  the  half  of  the  two  parts  (AD  +  DB)\ 
or,  y  L  equals  the  half  of  A  B\  or,  we  have — 


Adding  these  two  equations  together,  we  have — 


Now,  JD  +  DL  =  JL,  and  AD  +  DB  =  AB\    therefore, 


Thus  we  have  A  M  =  J  L.  From  each  of  these  equals 
take  J  M,  common  to  both,  then  the  remainders,  A  J  and 
ML,  will  be  equal  ;  therefore,  A  J  =  C  F. 

We  have  also  MB  =  J  L.  From  each  of  these  equals 
take  ML,  common  to  both,  and  the  remainders,  J  M  and 
L  B,  will  be  equal  ;  therefore,  L  B  =  E  C.  As  was  above 
shown  — 

RiAy'iiP-iL.9. 


TO    FIND   A   FOURTH    PROPORTIONAL.  377 

Substituting  for  A  J  and  LB  their  values,  as  just  found, 
we  have— 

R  :  CF  :  :  P  :  EC', 

from  which  we  have  (Art.  373)  — 

Px  CF=  R  x  E  C. 


Thus  it  is  demonstrated  that  the  product  of  one  weight  into 
its  arm  of  leverage,  is  equal  to  the  product  of  the  other 
weight  into  its  arm  of  leverage  :  a  proposition  which  is 
known  <is  the  law  of  the  lever. 

378.  —  Any  One  of  Four  Proportionals  may  be  Found. 

—  Any  three  of  four  proportionals  being  given,  the  fourth 
may  be  found  ;  for  either  one  of  the  four  factors  may  be 
made  to  stand  alone  ;  thus,  taking  the  equation  of  the  last 
article,  if  we  divide  both  members  by  CF  (Art.  371),  we 
have  — 

Px  CF'_RxEC 
C.F  CF    ' 

In  the  first  member  C  F,  in  both  numerator  and  denominator, 
cancel  each  other  (Art.  371),  therefore— 


so  likewise  we  may  obtain— 

Px  CF 


SECTION    VIII.— FRACTIONS. 

379. — A  Fraction  Defined. — As  a  fracture  is  a  break  or 
division  into  parts,  so  a  fraction  is  literally  a  piece  broken  off; 
a  part  of  the  whole. 

The  figures  which  are  generally  used  to  express  a  frac- 
tion show  what  portion  of  the  whole,  or  of  an  integer,  the 
fraction  is :  for  example,  let  the  line  A  B,  (Fig.  274),  be  divided 
into  five  equal  parts,  then  the  line  A  C,  containing  three  of 
those  parts,  will  be  three  fifths  of  the  whole  line  A  B,  and 

»  3 

may  be  expressed  by  the  figures  3  and   5,  placed  thus,  — , 

which  is  known  as  a  fraction  and  is  read,  three  fifths.  The 
number  5  below  the  line  deno'tes  the  number  of  parts  into 
which  an  integer  or  unit,  A  B,  is  supposed  to  be  divided  ;  it 


riii 1 1 

A       D      E      C  B 

FIG.  274. 

is  therefore  called  the  denominator,  and  expresses  th3  denom- 
ination or  kind,  whether  fifths,  sixths,  ninths,  or  any  number, 
into  which  a  unit  is  supposed  to  be  divided.  The  number 
3  above  the  line,  denoting  the  number  of  parts  contained  in 
the  fraction,  is  termed  the  numerator,  and  expresses  the 
number  of  parts  taken,  as  2,  3,  4,  or  any  other  number. 

380. — Graphical  Repre§entation  of  Fractions  :  Effect 
of  Multiplication. — In  Fig.  275,  let  the  line  A  B  be  di- 
vided into  three  equal  parts ;  the  line  CD  into  six  equal 
parts;  the  line  EF  into  nine  equal  parts;  the  line  G H  into 
twelve  equal  parts,  and  the  line  y A' into  fifteen  equal  parts. 
The  lines  AB,  CD,  EF,GH,  and  J K,  being  all  of  equal 
length. 


FRACTIONS   ILLUSTRATED. 


379 


Then  the  parts  of  these  lines,  A  L,  CM,  EN,  etc.,  may 

be  expressed   respectively  by  the  fractions-,^,-,  -  and  --. 

369   12  15 

In  each  case  the  figure  below  the  line,  as,  3,  6,  9,  12,  or  15, 
expresses  the  number  of  parts  into  which  the  whole  is  di- 
vided, and  the  figure  above  the  line,  as  1,2,  3,  4,  or  5,  the 


L 

M 

N 

D 

P 

FIG.  275. 

number  of  the  parts  taken  ;  and,  as  the  lines  A  Z,  CM,  EN, 
etc.,  are  all  equal  to  each  other,  therefore  these  fractions  are 
all  equal  to  each  other.  If  the  numerator  and  denominator 
of  the  first  fraction  be  each  multiplied  by  2,  the  products 
will  equal  the  numerator  and  denominator  of  the  second 
fraction ;  thus — 

1X2  =  2 
3X2  =  6' 


so,  also, 

and 

and 


I><  3  ~3. 
3x3=9' 

i_x  4=  4^ 
3  x4=  12 

ix  5  =_i_ 

3  x  5  =-15 


Thus  it  is  shown  that  when  the  numerator  and  denomi- 
nator of  a  fraction  are  each  multiplied  by  the  same  factor, 
the  product  forms  a  new  fraction  which  is  of  equal  value 
with  the  original. 

In  like  manner  we  have,  |,  — ,  A  --,  etc.,  each  equal  to 

o    12    It)   2O 

one  fourth;    and  which  may  be  found  by  multiplying' the 

numerator  and  denominator  of  -  successively  by  2,  3, 4,  5,  etc. 

4 


380  FRACTIONS.   • 

381. — Form  of  Fraction  Changed  by  Division. — By    an 

operation  the  reverse  of  that  in  the  last  article,  we  may  re- 
duce several  equal  fractions  to  one  of  equal  value.  Thus,  if 
in  each  we  divide  the  numerator  and  denominator  by  the 
same  number,  we  reduce  it  to  a  fraction  of  equal  value,  but 
with  smaller  factors. 

For  example,  taking  the  fractions  of  the  last  article,  f ,  f, 
iV  xV>  let  eacn  De  divided  by  a  number  which  will  divide 
both  numerator  and  denominator  without  a  remainder.* 

Thus,  ^"^2==I  1~"~3  =  l 

6  +  2*3'         9-^3  =  3" 

_4/r-4=  l         J_^-5  =1 
12-4=3'        15-^5  =  3* 

As  these  fractions  are  shown  (Art.  380)  to  be  equal,  and 
as  the  operation  of  dividing  each  factor  by  a  common  num- 
ber produces  quotients  which  in  each  case  form  the  same 
fraction,  -J-,  we  therefore  conclude  that  the  numerator  and 
denominator  of  a  fraction  may  be  divided  by  a  common 
number  without  changing  the  value  of  the  fraction. 

382. — Improper  Fractions. — The  fractions  f ,  ^,  ~,  etc., 
all  fractions  which  have  the  numerator  larger  than  the  de- 
nominator are  termed  improper  fractions.  They  are  not  im- 
proper arithmetically,  but  they  are  so  named  because  it  is  an 
improper  use  of  language  to  call  that  &part  which  is  greater 
than  the  whole. 

As  expressions  of  this  kind,  however,  are  sabject  to  the 
same  rules  as  those  which  are  fractions  proper,  it  is  custom- 
ary to  include  them  all  under  the  technical  term  of  fractions. 
Expressions  like  these — all  expressions  in  which  one  number 
is  separated  by  a  'horizontal  line  from  another  number  below 
it,  or  one  set  of  numbers  is  thus  separated  from  another  set 
below  it — may  be  called  fractions,  and  are  always  to  be  un- 
derstood as  indicating  division,  or  that  the  quantity  above 
the  line  is  to  be  divided  by  the  quantity  below  the  line. 


Division  is  indicated  by  this  sign  -r-,  which  is  read  "divided  by." 


IMPROPER  FRACTIONS.  381 

Q     17     2A    3x8x4    17x82 

Thus,  z>  — >  — *       ^      -»  >  etc.,  are  all  fractions,  tech- 

nically, although  each  may  be  greater  than  unity.  And  it  is 
understood  in  each  case  that  the  operation  of  division  is  re- 
quired. Thus,  -  =  3,  --  —  8,  —  —  =  4.  When  the  divis- 

33  %    , 

ion  cannot  be  made  without  a  remainder,  then  the  fraction, 
by  cutting  the  numerator  into  two,  may  be  separated  into  two 
parts,  one  of  which  may  be  exactly  divided,  and  the  other 

will  be  a  fraction  proper.     Thus,  the  fraction  -~  is  equal  to 

1 —  (for  15  +  2  —  17);  and    since  —   equals    3,  therefore, 

17          15       2  22 

—  =—  +  -  =  3  +  -  =  3-      So,  likewise,  the  fraction 

17x82  __  1394 :=i375+J9_.  .J_9_.         J_9_ 

125          125         125      125  125  125' 

383. — Reduction  of  Mixed  Numbers  to  Fractions — By 

an  operation  the  reverse  ot  that  in  the  last  article,  a  given 
mixed  number  (a  whole  number  and  fraction)  can  be  put 
into  the  form  of  an  improper  fraction. 

This  is  done  by  multiplying  the  whole  number  by  the  de- 
nominator of  the  fraction,  the  product  being  the  numerator 
of  a  fraction  equal  in  value  to  the  whole  number  ;  the  de- 
nominator of  this  fraction  being  the  same  as  that  of  the  given 
fraction.  The  numerator  of  this  fraction  being  added  to  the 
numerator  of  the  given  fraction,  the  sum  will  be  the  numera- 
tor of  the  required  improper  fraction,  the  denominator  of 
which  is  the  same  as  that  of  the  given  fraction.  For  example, 
the  required  numerator  for — 

2  J,  is  2  x  3  +  I  —  7.     So  2-3-  =  -J. 

2j,  is    2  X  4  +  I    =  9.       So  2\  =  f. 

3i  is  3  x  5  +2  =  17.  So3f  =  $. 

384. — Division  Indicated  by  the  Factors  put  as  a  Frac- 
tion.— Factors  placed  in  the  form  of  a  fraction  as  — ,  -,  — -  or 


382  FRACTIONS. 

— -  indicate  division  (Art.  382) ;  the  denominator  (the  fac- 
tor below  the  line)  being  the  divisor,  and  the  numerator 
(the  factor  above  the  line)  the  dividend,  while  the  value  of 

the  fraction  is  the  quotient.     Thus  of  the  fraction,  —  =  20, 

9  41 

41  is  the  divisor,  820  the  dividend,  and  20  the  quotient. 
From  this  we  learn  that  division  may  always  be  indicated 
by  placing  the  factors  in  the  form  of  a  fraction,  so  that  the 
divisor  shall  form  the  denominator  and  the  dividend  the  nu- 
merator. 


385.  —  Addition  of  Fractions  having  Like  Denomina- 
tors —  Let  it  be  required  to  add  the  fractions  -  and  -.  By 

referring  to  Art.  379  we  see  that  ^4  D  (Fig.  274),  is  one  of  the 
five  parts  into  which  the  whole  line  A  B  is  divided  ;  it  is, 

therefore,  —  .     We  also  see  that  D  C  contains  two  of  the  five 

2 

parts  ;  it  is,  therefore,  -.  We  also  see  that  AD  +  D  C  '  —  A  C, 
which  contains  three  of  the  five  parts,  or  A  C  =  —  of  A  B. 

12  3 

We  therefore  conclude  that  —  +  —  =  —  .     In  this  operation  it 

is  seen  that  the  denominator  is  not  changed,  and  that  the 
resultant  fraction  has  for  a  numerator  a  number  equal  to  the 
sum  of  the  numerators  of  the  fractions  which  were  required 
to  be  added. 

By  this  it  is  shown  that  to  add  fractions  we  simply  take 
the  sum  of  the  numerators  for  the  new  numerator,  making  the 
denominator  of  the  resultant  fraction  the  same  as  that  of  the 
fractions  to  be  added.  For  example  :  What  is  the  sum  of  the 

fractions  —  ,  —  and  -  ?  Here  we  have  14-3+4  —  8  for  the 
numerator,  therefore  — 


999 


SUBTRACTION   AND   ADDITION   OF   FRACTIONS.  383 

386.— Subtraction  of  Fractions  Of  Like  Denominators.— 

Subtraction  is  the  reverse  of  addition ;  therefore,  to  sub- 
tract fractions  a  reverse  operation  is  required  to  that  had  in 
the  process  of  addition  ;  or  simply  to  subtract  instead  of 
adding. 

2  ^ 

For  example,  if  -  be  required  to  be  su>tracted  from  — 
we  have— 

UNIVERSITY 

5     5  " 


By  reference  to  Fig.  274  an  exemplification  of^tkis-wiit' 

•?  2  T 

seen  where  we   have  A  C  =  — ,  A  E  =  — ,  and  E  C  =  — ,  and 
we  have — 


3  _2  =£ 

5  5       5' 

We  therefore  have  this  rule  for  the  substraction  of  frac- 
tions :  Subtract  the  less  from  the  greater  numerator  ;  the  remain- 
der will  be  the  numerator  of  the  required  fraction.  The  denom- 
inator to  be  the  same  as  that  of  the  given  fractions. 

387.— Dissimilar  Denominators  Equalized. — The  rules 
just  given  for  the  addition  and  subtraction  of  fractions  re- 
quire that  the  given  fractions  have  like  denominators. 
When  the  denominators  are  unlike  it  is  required,  before  add- 
ing or  substracting,  that  the  fractions  be  modified  so  as  to 
make  the  denominators  equal.  For  example :  Let  it  be  re- 
quired to  find  the  sum  of  -  and  -.  By  reference  to  Fig. 

2  6 

275,  we  find  that  —on  line  A  B  is  equal   to  -  on    line   E  F. 

These  being   equal,  we  may    therefore  substitute  —  for  -. 
Then  we  have — 

6  2_  _    8 
9  +  9  "  9 


384  FRACTIONS. 

Now,  it  will  be  seen  that  the  fraction  -  may  be  had  by  mul- 
tiplying both  numerator  and  denominator  of  the  given  frac- 

2     ,  2X^  =  6 

tion-  by  3,  for  3  x  -  =  -; 

and  we  have  seen  (Art.  380)  that  this  operation  does  not 
change  the  value  of  the  fraction.  From  this  we  learn  that 

the  denominators  may  be  made  equal  by  multiplying  the  smaller 
denominator  and  Its  numerator  by  any  number  which  will  effect 
such  a  result. 

For  example  :          ^-+-—  =  —  +  —  =  -^-; 
27         14        7         21 


3    ,3         7         12       4        7        23  7 

4+I7+T6  =:T6+7^+7^=^z    '76' 

In  this  example  the  second  fraction  is  changed  by  multiply- 
ing by  i  j. 

388.  —  Reduction  of  Fraction§  to  tlieir  Lowest  Terms.  — 

The  process  resorted  to  in  the  last  article  to  equalize  the 
denominators,  is  not  always  successful.  What  is  needed  for 
a  common  denominator  is  to  find  the  smallest  number 
which  shall  be  divisible  by  each  of  the  given  denominators. 
Before  seeking  this  number,  let  each  given  fraction  be 
reduced  to  its  lowest  terms,  by  dividing  each  factor  by  a 

common  number.    For  example:  —  may,  by  dividing  by  5, 

be  reduced  to  —  ,  which  is  its  equivalent.     So,  also,  —  ,  by  di- 
3  2o 

viding  by  7,  is  reduced  to  —  ,  its  lowest  terms. 

389.  —  Lea§t  Common  Denominator.  —  To  find  the  least 
common  denominator  ^V&w  the  several  fractions  in  the  order 
of  their  denominators,  increasing  toward  the  right.     If  the 
largest  denominator  be  not  divisible  by  each  of  the  others, 
double  it  ;  if  the  division  cannot  now  be  performed,  treble 


LEAST  COMMON  DENOMINATOR.  385 

it,  and  so  proceed  until  it  is  multiplied  by  some  number 
which  will  make  it  divisible  by  each  of  the  other  denomina- 
tors. This  number  multiplied  by  the  largest  denominator  will  be 
the  least  common  denominator.  To  raise  the  denominator  of 
each  fraction  to  this,  divide  the  common  denominator  by  the  de- 
nominator of  one  of  the  fractions,  the  quotient  will  be  the 
number  by  which  that  fraction  is  to  be  multiplied,  both 
numerator  and  denominator,  and  so  proceed  with  each  frac- 
tion. For  example :  What  is  the  sum  of  the  fractions 

-,  -,  — ,  -g?     One  of  these,  — ,  may  be  reduced,  by  divid- 
ing by  2,  to  ^.     Therefore,  the  series  is  -,  -,  -|»  ~.    On  trial 
o  2    A.    \)    o 

we  find  that  8,  the  largest  denominator,  is  divisible  by  the 
first  and  by  the  second,  but  not  by  the  third,  therefore  the 
largest  denominator  is  to  be  doubled:  2x8=  16.  This  is 
not  yet  divisible  by  the  third  ;  therefore  3  x  8  =  24.  This 
now  is  divisible  by  the  third  as  well  as  by  the  first  and  the 
second  ;  24  is  therefore  the  least  common  denominator. 

Now  dividing  24  by  2,  the  first  denominator,  the  quotient 
12  is  the  factor  by  which  the  terms  of  the  first  fraction  are 

to  be  raised,  or,  -  ~  — -.       For  the   second    we   have 

24-5-4  —  6,  and  -     ^  — .    For  the  third  we  have  24  -*-  6  = 
4x0  =  24 

4,    and    ~X      7 — J    an<^    f°r    tne    fourth,    24-^-8  =  3,    and 
o  x  4  —  24 

7  X  3  — -21 

^        ~_— .     Thus  the  fractions  in  their  reduced  form  are  : 
12       18       20       21       7i         23 

I        I         I        .  •  ...        n ^ 

24  24  24  24  ~~       24     ~~  24* 

390. — Lea§t  Common  Denominator  Again. — When  the 
denominators  are  not  divisible  by  one  another,  then  to  ob- 
tain a  common  denominator,  it  is  requisite  to  multiply  to- 
gether all  of  the  denominators  which  will  not  divide  any  of  the 
other  denominators.  For  example :  What  is  the  sum  of  the 

fractions  -,  -,  -,  and  -? 


386 


FRACTIONS. 


In  this  case  the  first  denominator  will  divide  the  last,  but 
the  others  are  prime  to  each  other.  Therefore,  for  the 
common  denominator,  multiply,  together  all  but  the  first  ; 
or  — 

5x7x9  =  315  the  common  denominator  ; 
and  — 

315  _:_  3  —  105,  common  factor  for  the  first  fraction  ; 
315  -=-  5  =  63,    common  factor  for  the  second  fraction  ; 
315  _i_  7  •=.  45,    common  factor  for  the  third;    . 
315-5-9  =  35,    common  factor  for  the  fourth. 

And,  then  — 

« 

i  x  105  =  105  t    2  x  63  =  126      3  x  45  =  135  ^  4  x  35  =  140 

1  x  105  =  315  '  "5  x  63  =  3"i~5  '  7  x  45  -  3^5  '  9  x  35  =      " 


105      126      135      140  _  506 
+  315  "h  3i5  *  3i5  ~~  3^5 


191 


39(.—  Fraction§  multiplied  Graphically.—  Let   A  B  CD 

(Fig.  276)  be  a  rectangle  of  equal  sides,  or  A  B  equal  A  C 
and  each  equal  one  foot.     Then  A  B  multiplied  by  A  C  will 


G 

C                H               ( 
FIG.  276. 

equal  the  area  A  B  CD,  or  i  x  i  =  i  square  foot.  Let  the 
line  E  F  be  parallel  with  A  B,  and  midway  between  -A  B  and 
CD.  Then  A  B  x  A  £  equals  half  the  area  of  A  B  CD,  or 
i  x  J  =  -J.  Again  ;  let  G  H  be  parallel  with  E  C,  and  mid- 
way between  E  C  and  FD.  Then  E  G  x  E  C  =  i  x  £  equals 
the  area  E  G  C  H,  which  is  equal  to  a  quarter  of  the  area 


MULTIPLICATION   OF  FRACTIONS. 


387 


A  B  C D;  or  %  x  %  =  J;  which  is  a  quarter  of  the  superficial 
area. 

The  product  here  obtained  is  less  than  either  of  the 
factors  producing  it.  It  must  be  remembered,  however, 
that  while  the  factors  represent  lines,  the  product  represents 
superficial  area.  The  correctness  of  the  result  may  be 
recognized  by  an  inspection  of  the  diagram. 

392. — Fraction§  multiplied  Graphically.  —  In  Fig.  277 
let  A  B  equal  8  feet  and  A  C  equal  5  feet ;  then  the  rect- 


G 


FIG.  277. 


angle  A  B  CD  contains  5  x  8  =  40  feet.     The  interior  lines 
divide   the   space    included   within   A  B  CD  into  40   equal 

squares  of  one  foot  each.     Let  A  E  equal  3  feet  or  -  of  A  C. 
Let  A  G  equal   7  feet  or  ~  of  A  B.     Then   the   rectangle 

3         7  21 

E  F  A  G  contains  — x  '—  —  — ,  or  twenty-one  fortieths  of  the 
5      04° 

<+  *j 

whole  area  A  B  CD.    Thus,  while  the  factor  fractions  --  and  -^ 

5          o 

represent  lines,  it  is  shown  that  the  product  fraction  —  rep- 


40 


21    . 


resents  surface.     Thus  —  is  a  fraction,  E  FA  G,  of  the  whole 

40 

surface,  CDAB. 

393. — Rule  for  Miitiplication  of  Fraction*,  and  Exam- 
ple.— In  the  example  given  in  the  last  article  it  will  be  ob- 


388  FRACTIONS. 

served  that  the  product  of  the  denominators  of  the  two 
given  fractions  equals  the  area  of  the  whole  figure  (A  B  C  D\ 
while  the  product  of  the  numerators  equals  the  area  of  the 
rectangle  (E 'FA  G),  the  sides  of  which  are  equal  respec- 
tively to  the  given  fractions.  From  this  we  obtain  for  the 
product  of  fractions  this— 

RULE. — Multiply  together  the  denominators  for  the  new  de- 
nominator, and  the  numerators  for  a  new  numerator. 

j  ?  j 

For  example:  what  is  the  product  of   —  and    —  ?     Here 

we  have  20x21—420  for  the  new  denominator,  and 
7  x  13  =  91  for  the  new  numerator;  therefore  the  product 
of— 

il  x_7  =    _£L. 

21         2O  420 ' 

or,  of  a  rectangular  area  divided  one  way  into  20  parts  and 
the  other  way  into  21  parts,  thus  containing  420  rectangles, 

1 3  7 

the  product  of  the  two  fractions  —  and  - —  is  equal  to  91  of 

these  rectangles,  or  —  -  of  the  whole. 

394. — Fraction§  Divided  Graphically. —  Division  is  the 
reverse  of  multiplication ;  or,  while  multiplication  requires 
the  product  of  two  given  factors,  division  requires  one  of 
the  factors  when  the  other  and  the  product  are  given.  Or 
(referring  to  Fig.  277)  in  division  we  have  the  area  of  the 
rectangle,  E  FA  G,  and  one  side,  E  A;  given,  to  find  the 
other  side,  A  G. 

Now  it  is  required  to  find  the  number  of  times  E  A  is 
contained  in  E  FA  G.  By  inspection  of  the  figure  we  per- 
ceive the  answer  to  be,  A  G  times ;  for  E  A  xAG  —  EFA  G, 

2  I 

the  given  area.  Or,  when  E  A  F  G  is  given  as  —  and  E  A 
as  -,  we  have  as  the  given  problem — 

ILL.! 

40    '    5* 


DIVISION   OF  FRACTIONS.  389 

Since  division  is  the  reverse  of  multiplication,  instead  of 
multiplying  we  divide  the  factors,  and  have— 

21  -i-  3  =  7 
40  -r-  5  "    8* 

Thus,  to  divide  one  fraction  by  another,  for  the  numerator  of 
the  required  factor,  divide  the  numerator  of  t/ie  product  by  the 
numerator  of  the  given  factor,  and  for  the  denominator  of  the 
required  factor  divide  the  denominator  of  the  product  by  the 
denominator  of  the  given  factor.  For  example  : 

10  .      2  5 

Divide  ^~  by  -.     Answer,  —  . 

o  Q  >|  /•» 

Divide  -  -  by  —  .     Answer,  —  . 

395.  —  Rule  for  Division  of  Fractions.  —  The   rule  just 
given  does  not  work  well  when  the  factors  are  not  commen- 

5        2 
surable.     For  example,  if  it  be  required  to  divide  —  by  —  we 

have  by  the  above  rule— 


7-9    "  7  ' 
9. 

Producing  fractional  numerators  and  denominators  for  the 
resulting  fraction,  which  require  modification  in  order  to 
reach  those  composed  only  of  whole  numbers.  If  the  nu- 
merators, 5  and  7,  of  this  compound  fraction  be  multiplied 
by  9  (the  denominator  of  the  denominator  fraction),  or  the 
compound  fraction  by  9,  we  shall  have  — 


5  ><9 


390  FRACTIONS. 

And,  if  these  be  again  multiplied  by  2  (the  denominator  of 
the  numerator  fraction),  we  shall  have  — 


5  X9 

2 


—  X  2  = 


7x9  7x9x2 

~9~  ~9~ 

Like  figures  above  and  below  in  each  fraction  cancel  each 
other  (Art.  371),  therefore,  the  result  reduces  to — 

5  x  9 

7x2' 

in  which  we  find  the  factors  of  the  two  original  fractions. 
In  one  fraction  —  we  have  the  factors  in  position  as  given, 

but  in  the  other  —  they  are  inverted.    The  fraction  in  which. 

the  factors  are  inverted  is  the  divisor.     Hence,  for  division 
of  fractions,  we  have  this — 

RULE. — Invert  the  factors  of  the  divisor,  and  then,  as  in 
multiplication,  multiply  the  numerators  together  for  the  numera- 
tor of  the  required  fraction,  and  the  denominators  for  the  de- 
nominator of  the  required  fraction. 

c  2 

Thus,  as  before,  if  -  is  required  to  be  divided  by  -,  we 

have — 

ix9  „  45 
7x2        14' 

And,  to  divide  —  by  — ,  we  have — 

23  x  9  _  207 
47  x  7       329 

2£  8 

Again,  to  divide  —  by  — ,  we  have — 

2$  x  9  _  225  =  _25  =  _s 
45  x  8  ""  360    "40    "8" 


CANCELLING   IN  ALGEBRA.  39! 

This  last  example  has  two  factors,  9  and  45,  one  of  which 

measures  the  other  ;  also,  the  first  fraction  -  -  is  not  in  its 

45 

lowest  terms;  when  reduced  it  is  — .     The  question,  there- 
i  fore,  may  be  stated  thus : 

5  x  9       1 . 
9  x  8  ""  8  ' 

for  the  two  9*8  cancel  each  other. 


SECTION  IX.— ALGEBRA. 

396. — Algebra  Defined. — It  occurs  sometimes  that  a 
student  familiar  only  with  computation  by  numerals  is 
needlessly  puzzled,  in  approaching  the  subject  of  Algebra, 
to  comprehend  how  it  is  possible  to  multiply  letters  together, 
or  to  divide  them.  To  remove  this  difficulty,  it  may  be  suf- 
ficient for  them  to  learn  that  their  perplexity  arises  from  a 
misunderstanding  in  supposing  the  letters  themselves  are 
ever  multiplied  or  divided.  It  is  true  that  in  treatises  on 
the  subject  it  is  usual  to  speak  as  though  these  operations 
were  actually  performed  upon  the  letters.  It  is  always  un- 
derstood, however,  that  it  is  not  the  letters,  but  the  quan- 
tities represented  by  the  letters,  which  are  to  be  multiplied 
or  divided. 

For  example,  in  Art.  361  it  is  shown,  in  comparing  similar 
sides  of  homologous  triangles,  that  the  bases  of  the  two  tri- 
angles are  to  each  other  as  the  corresponding  sides,  or, 
referring  to  Fig.  269,  we  have  C E  :  A  E  :  :  D  E  :  B  E. 
Now,  let  the  two  bases  C  E  and  A  E  be  represented  respec- 
tively by  a  and  b,  and  the  two  corresponding  sides  D  E  and 
B  E  by  c  and  d  respectively  ;  or,  for— 

CE  :  AE  :  :  DE  :  BE, 
put — 

a  :  b  :  :  c  :  d\ 

and,  by  Art.  373,  we  have — 

b  x  c  =  a  x  d, 
which  may  be  written— 

be  =  ad\ 

for  x,  the  sign  for  multiplication,  is  not  needed  between  let- 
ters, as  it  is  between  numeral  factors.     The  operation  of 


APPLICATION  OF  ALGEBRA.  393 

multiplication  is  always  understood  when  letters  are  placed 
side  by  side. 

Now,  here  we  have  an  equation  in  which,  as  usually  read, 
we  have  the  product  of  b  and  c  equal  to  the  product  of  a 
and  d.  But  the  meaning  is  that  the  product  of  the  quantities 
represented  by  b  and  c  is  equal  to  the  product  of  the  quan- 
tities represented  by  a  and  d,  and  that  this  equation  is  in- 
tended to  represent  the  relation  subsisting  between  the  four 
proportionals,  C  £,  A  E,  D  E,  and  BE,  of  Fig.  269.  In  order 
to  secure  greater  conciseness  and  clearness,  the  four  small 
letters  are  substituted  for  the  four  pair  of  capital  letters, 
which  are  used  to  indicate  the  lines  of  the  figures  referred  to. 

397.  —  Example  :  Application.  —  It  was  shown  in  the  last 
article  that  the  four  letters  a,  £,  c,  and  d  represent  the  cor- 
responding sides  of  the  two  triangles  of  Fig..2§g,  and  that  — 

b  c  =  a  d. 

Now,  let  each  member  of  this  equation  be  divided  by  a,  then 
(Art.  371)- 


If  now  the  dimensions  of  the  three  sides  represented  by  a, 
b,  and  c  are  known,  and  it  is  required  to  ascertain  from  these 
the  length  of  the  side  represented  by  d,  let  the  three  given 
dimensions  be  severally  substituted  for  the  letters  repre- 
senting them.  For  example,  let  a  =  40  feet  ;  b  =  52  feet, 
and  c  =  45  feet  ;  then— 


be       52  x  45 

d  =  —  =  —      -  =  -~  =  58 
a  40  40 

The  quantities  being  here  substituted  for  the  letters  ;  we  have 
but  to  perform  the  arithmetical  processes  indicated  to  obtain 
the  arithmetical  value  of  d.  From  this  example  it  is  seen 
that  before  any  practical  use  can  be  made  of  an  algebraical 
formula  in  computing  dimensions,  it  is  requisite  to  substitute 
numerals  for  the  letters  and  actually  perform  arithmetically 
such  operations  as  are  only  indicated  by  the  letters. 


394  ALGEBRA. 

398. — Algebra   Useful   in    Constructing    Rules. — In    all 

problems  to  be  solved  there  are  certain  conditions  or  quan- 
tities given,  by  means  of  which  an  unknown  quantity  is  to 
be  evolved.  For  example,  in  the  problem  in  Art.  397,  there 
were  three  certain  lines  given  to  find  a  fourth,  based  upon 
the  condition  that  the  four  lines  were  four  proportionals. 
Now,  it  has  been  found  that  the  relation  between  quantities 
and  the  conditions  of  a  question  can  better  be  stated  by  let- 
ters than  by  numerals ;  and  it  is  the  office  of  algebra  to 
present  by  letters  a  concise  statement  of  a  question,  and  by 
certain  processes  of  comparison,  substitution  and  elimina- 
tion, to  condense  the  statement  to  its  smallest  compass,  and 
at  last  to  present  it  in  a  formula  or  rule,  which  exhibits  the 
known  quantities  on  one  side  as  equal  to  the  unknown  on 
the  other  side.  Here  algebra  ends,  at  the  completion  of  the 
rule.  To  use  the  rule  is  the  office  of  arithmetic.  For,  in 
using  the  rule,  each  quantity  in  numerals  must  be  substi- 
tuted for  the  letter  representing  it,  and  the  arithmetical 
processes  indicated  performed,  as  was  done  in  Art.  397. 

399.— Algebraic  Rule§  are  General. — One  advantage 
derived  from  algebra  is  that  the  rules  made  are  general 
in  their  application,  For  example,  the  rule  of  Art.  397, 

—  =  d,  is  applicable  to  all  cases  of  homologous  triangles, 

however  they  may  differ  in  size  or  shape  from  those  given  in 
Fig.  269 — and  not  only  this,  but  it  is  also  applicable  in  all 
cases  where  four  quantities  are  in  proportion  so  as  to  con- 
stitute four  proportionals.  For  example,  the  case  of  the 
four  proportionals  constituting  the  arms  of  a  lever  and  the 
weights  attached  (Arts.  375-378).  For,  taking  the  rela- 
tion as  expressed  in  Art.  377 — 

PxCF=  RxEC, 

we  may  substitute  for  C  F  the  letter  n,  and  for  E  C  the  letter 
m,  then  m  will  represent  the  arm  of  the  lever  E  C  (Fig.  262), 
aid  n  *he  arm  of  the  lever  F  C.  Then  we  have— 


SYMBOLS   CHOSEN  AT  PLEASURE.  395 

and  from  this,  dividing  by  n  (/Irt.  372),  we  have  — 


or,  dividing  by  m,  we  have  — 

(in.)    - 

v        ' 


m 


which  is  a  rule  for  computing  the  weight  of  R,  when  P  and 
the  two  arms  of  leverage,  m  and  n,  are  known.  For  example, 
let  the  weight  represented  by  P  be  1200  pounds,  the  length 
of  the  arm  m  be  4  feet,  and  that  of  n  be  8  feet,  then  we  have— 

Pn      1  200  x  8 

R  =  -  -  =  -  —  =  2400  pounds. 
m  4 

Pn 
This   rule,   R  =.  —  -,   is   precisely   like    that    in   Art.    397  — 

—  —  d  —  in  which  three  quantities  are  given  to  find  a  fourth, 
the  four  constituting  a  set  of  four  proportionals. 

400.  —  Symbols  €ho§en  at  Pleasure.  —  The  particular 
letter  assigned  to  represent  a  particular  quantity  is  a  matter 
of  no  consequence.  Any  letter  at  will  may  be  taken  ;  but 
when  taken,  it  must  be  firmly  adhered  to  to  represent  that  par- 
ticular quantity,  throughout  all  the  modifications  which  may 
be  requisite  in  condensing  the  statement  into  which  it  enters 
into  a  formula  for  use.  For  example,  the  two  rules  named  in 
Art.  399  are  precisely  alike  —  three  quantities  given  to  find  a 
fourth  —  yet  they  are  represented  by  different  letters.  In  one, 
R  and  P  represent  the  two  weights,  and  m  and  ft  the  arms  of 
leverage  at  which  they  act  ;  while  in  the  other  the  letters 
a,  b,c,  and  ^represent  severally  the  four  lines  which  constitute 
two  similar  sides  of  two  homologous  triangles.  The  two 
rules  are  alike  in  working,  and  they  might  have  been  con- 
stituted with  the  same  letters.  And  instead  of  the  letters 
chosen  any  others  might  have  been  taken,  which  con- 
venience or  mere  caprice  might  have  dictated.  In  some 


ALGEBRA. 

questions  it  is  usual  to  put  the  first  letters,  as  a,  b,  c,  etc.,  to 
represent  known  quantities,  and  the  last  letters,  as  x,  y,  z, 
for  the  quantities  sought.  In  works  on  the  strength  of 
materials  it  is  customary  to  represent  weights  by  capital 
letters,  as  P,  R,  U,  W,  etc.,  and  lines  or  linear  dimensions  by 
the  small  letters,  as  b,  d,  /,  for  the  breadth,  depth,  and  length, 
respectively,  of  a  beam.  Any  other  letters  may  be  put  to 
represent  these  quantities,  although  the  initial  letter  of  the 
word  serves  to  assist  the  memory  in  recognizing  the  partic- 
ular dimensions  intended. 

40  1.  —  Arithmetical  Processe§  Indicated  by   Sign§.  —  In 

algebra,  the  four  processes  of  addition,  subtraction,  multi- 
plication, and  division,  are  frequently  required  ;  and  when 
the  required  process  cannot  be  actually  performed  upon  the 
letters  themselves,  a  certain  method  has  been  adopted  by 
which  the  process  is  indicated.  For  example,  in  additon, 
when  it  is  required  to  add  a  to  b,  the  two  letters  cannot  be 
intermingled  as  numerals  may  be,  and  their  sum  presented  ; 
but  the  process  of  addition  is  simply  indicated  by  placing 
between  the  two  letters  this  sign,  +,  which  is  called  plus, 
meaning  added  to  ;  therefore,  to  add  a  to  b  we  have  — 


which  is  read  a  plus  b,  or  the  sum  of  a  and  b.  When  the 
quantities  represented  b}^  a  and  b  are  substituted  for  them  — 
and  not  till  tHen  —  they  can  be  condensed  into  one  sum. 
For  example,  let  a  equal  4  and  b  equal  3,  then  for— 

a-\-b 
we  have  — 

4+3; 

and  we  may  at  once  write  their  sum  7,  instead  of  4  +  3. 

So,  likewise,  in  the  process  of  subtraction,  one  letter  can- 
not be  taken  from  another  letter  so  as  to  show  how  much  of 
this  other  letter  there  will  be  left  as  a  remainder  ;  but  the 
process  of  subtraction  can  be  indicated  by  a  sign,  as  this,  —  , 
which  is  called  minus,  less,  meaning  subtracted  from.  For 


ALGEBRAICAL   SIGNS.  397 

example,  let  it  be  required  to  subtract  b.  from  a.     To  do 
this  we  have  — 


which  is  read  a  minus  b,  and  when  the  values  of  a  and  b  are 
substituted  for  them,  we  have,  when  a  equals  4,  and  b 
equals  3— 

a-b, 
or  — 

4-3; 


and  now,  instead  of  4  —  3,  we  may  put  the  value  of  the 
which  is  unity,  or  i. 

The  algebraic  signs  most  frequently  used  are  as  follows  : 

+  ,//#.$•,  signifies  addition,  and  that  the  two  quantities  be- 
tween which  it  stands  are  to  be  added  together;  as 
a  +  b,  read  a  added  to  b. 

—  ,  minus,  signifies  subtraction,  or  that  of  the  two  quantities 
between  which  it  occurs>  the  latter  is  to  be  subtracted 
from  the  former  ;  as  a  —  b,  read  a  minus  b. 

X,  multiplied  by,  or  the  sign  of  multiplication.  It  denotes 
that  the  two  quantities  between  which  it  occurs  are  to 
be  multiplied  together  ;  as  a  x  b,  read  a  multiplied  by  b, 
'  or  a  times  b.  This  sign  is  usually  omitted  between 
symbols  or  letters,  and  is  then  understood,  as  a  b.  This 
has  the  same  meaning  as  a  x  b.  It  is  never  omitted 
between  arithmetical  numbers;  as  9x5,  read  nine 
times  five. 

-^,  divided  by,  or  the  sign  of  division,  and  denotes  that  of  the 
two  quantities  between  which  it  occurs,  the  former  is 
to  be  divided  by  the  latter;  as  a~b,  read  a  divided  by 
b.  Division  is  also  represented  thus  : 

-,  in  the  form  of  a  fraction.  This  signifies  that  a  is  to  be 
divided  by  b.  When  more  than  one  symbol  occurs 

above  or  below  the  line,  or  both,  as  --  ,   it    denotes 

that  the  product  of  the  symbols  above  the  line  is  to  be 
divided  by  the  product  of  those  below  the  line. 


39^  ALGEBRA. 

=  ,  is  equal  to,  or  sign  of  equality,  and  denotes  that  the 
quantity  or  quantities  on  its  left  are  equal  to  those  on 
its  right ;  as  a  —  b  =  c,  read  a  minus  b  is  equal  to  cy  or 
equals  c  ;  or,  9  —  5  =  4,  read  nine  minus  five  equals 
four.  This  sign,  together  with  the  symbols  on  each 
side  of  it,  when  spoken  of  as  a  whole,  is  called  an 
equation. 

a1  denotes  a  squared,  or  a  multiplied  by  a,  or  the  second 
power  of  a,  and 

a*  denotes  a  cubed,  or  a  multiplied  by  a  and  again  multi- 
plied by  a,  or  the  third  power  of  a.  The  small  figure, 
2,  3,  or  4,  etc.,  is  termed  the  index  or  exponent  of  the 
power.  It  indicates  how  many  times  the  symbol  is  to 
be  taken.  Thus,  d1  =  a  a,  a3  =  a  a  a,  a"  —  a  a  a  a. 

\/  is  the  radical  sign,  and  denotes  that  the  square  root  of  the 
quantity  following  it  is  to  be  extracted,  and 

I/  denotes  that  the  cube  root  of  the  quantity  following  it  is 
to  be  extracted.  Thus,  4/9  =  3,  and  V27  —  3-  The 
extraction  of  roots  is  also  denoted  by  a  fractional  in- 
dex or  exponent,  thus — 

a1/*  denotes  the  square  root  "of  a, 

a*  denotes  the  cube  root  of  a, 

a*  denotes  the  cube  root  of  the  square  of  a,  etc. 

402. — Example  in  Addition  and  Subtraction :  Cancel- 
ling.— Let  there  be  some  question  which  requires  a  state- 
ment to  represent  it,  like  this — 

a-i-d  =  c  —  b, 

which  indicates  that  if  the  quantity  represented  by  a  be 
added  to  the  quantity  represented  by  d,  the  sum  will  be 
equal  to  the  quantity  represented  by  c,  after  there  has  been 
subtracted  from  it  the  quantity  represented  by  b ;  or,  as  it  is 
usually  read,  a  plus  d  equals  c  minus  b ;  or  the  sum  of  a  and 
d  equals  the  difference  between  c  and  b.  For  illustration, 
take  in  place  of  these  four  letters,  in  the  order  they  stand, 
the  numerals  4,  2,  9,  3,  and  we  shall  havq  by  substitution  - 

a  +  d  —  c  —  b, 

4+2—9  —  3,  or  adding 

and  subtracting—  6  =  6. 


TRANSFERRING   SYMBOLS.  399 

If  it  be  required  to  add  to  each  member  of  the  equation 
the  quantity  represented  by  b,  this  will  not  interfere  with 
the  equality  of  the  members.  For  a  +  d  are  equal  to  c  —  d, 
and  if  to  each  of  these  two  equals  a  com-mon  quantity  be 
added,  the  sums  must  be  equal  ;  therefore  — 

a  +  d+  b  =  c  —  b-^b, 
or  by  numerals  — 

4  +  2  +  3  =  9-3  +  3, 
or  — 

9  =  9- 

It  will  be  observed  that  the  right  hand  member  contains 
the  quantity  —  b  and  +  b.  This  shows  that  the  quantity  b  is 
to  be  subtracted  and  then  added.  Now,  if  3  be  subtracted 
from  9,  the  remainder  will  be  6,  and  then  if  3  be  added,  the 
sum  will  be  9,  the  original  quantity.  Thus  it  is  seen  that 
when  in  the  same  member  of  an  equation  a  symbol  appears  as  a 
minus  quantity  and  also  as  a  plus  quantity,  the  two  cancel  each 
other,  and  may  be  omitted.  Therefore,  the  expression— 


b  =  c  —  b  +  b 
becomes  — 

a  +  d+  b  =  c. 

403.  —  Transferring  a  Symbol  to  the  Opposite  member. 

—  In  comparing,  in  the  last  article,  the  first  equation  with  the 
last,  it  will  be  seen  that  the  same  symbols  are  contained  in 
each,  but  differently  arranged  :  that  while  in  the  first  equa- 
tion b  appears  in  the  right  hand  member  and  with  a  minus 
or  negative  sign,  in  the  last  equation  it  appears  in  the  left 
hand  member  and  with  a  plus  or  positive  sign.  Thus  it  is 
seen  that  in  the  operation  performed  b  has  been  made  to 
pass  from  one  member  to  the  other,  but  in  its  passage  it  has 
been  changed.  A  similar  change  may  be  made  with  another 
of  the  symbols.  For  example,  from  the  last  equation,  let  d 
be  subtracted,  or  this  process  indicated,  thus  — 

a  +  d+b  —  d  =  c  —  d. 


400  ALGEBRA. 

The  plus  and  minus  d,  in  the  left  hand  member  cancel  each 
other,  therefore— 

a  +  b  =  c  —  d, 
or,  by  numerals  — 

4+3=9—  2- 
Reducing  — 

7  =  7. 

By  this  we  learn  that  any  quantity  (connected  by  +  or  —  ) 
may  be  passed  from  one  member  of  the  equation  to  the  other,  pro- 
vided the  sign  be  changed. 

404.  —  Signs  of  Symbols  to  be  Changed  when  they  are 
to  »e  Subtracted,  —  As  an  example  in  subtraction,  let  the 
quantities  represented  by  +  b  —  a  —f+  c,  be  taken  from  the 
quantities  represented  by  +  a+b  —  c—  f.  This  may  be 
written  — 

(+a  +  b  —  c  —f)  —  (+b  —  a—f+c), 


an  expression  showing  that  the  quantities  enclosed  within  the 
second  pair  of  parentheses  are  to  be  .subtracted  from  those 
included  within  the  first  pair.  Let  the  quantities  represent- 
ed in  the  first  pair  of  parentheses  for  convenience  be  repre- 
sented by  A  ,  or,  a  +  b  —  c  —f  —  A  .  Now,  by  the  terms  of  the 
problem,  we  are  required  to  subtract  from  A.  the  quantities 
enclosed  within  the  second  pair  of  parentheses.  To  do  this 
take  first  the  positive  quantity,  b,  and  subtract  it  or  indicate 
the  subtraction,  thus  — 

A-b; 

we  will  then  subtract  the  positive  quantity  c,  or  indicate  the 
subtraction,  thus  — 

A-b-c. 

We  have  yet  to  subtract  —  a  and  —  /,  two  negative  quanti- 
ties. 

The  method  by  which  this  can  be  accomplished  may  be 
discovered  by  considering  the  requirements  of  the  problem. 
The  plus  quantities  b  and  £,  before  being  subtracted  from  A, 
were  required  to  have  the  two  negative  quantities  #  and  /de- 


THE   OPERATION  TESTED.  40  1 

ducted  from  them.  It  is  evident,  therefore,  that  in  subtract- 
ing b  and  c,  before  this  deduction  was  made,  too  much  has 
been  taken  from  A,  and  that  the  excess  taken  is  equal  to  the 
sum  of  a  and  /.  To  correct  the  error,  therefore,  it  is  neces- 
sary to  add  just  the  amount  of  the  excess,  or  to  add  the  sum 
of  a  and/,  or  annex  them  by  the  plus  sign,  thus  — 

A  —  b  —  c  +  a+f. 

To  test  the  correctness  of  the  operation  as  here  performed, 
let  numerals  be  substituted  for  the  symbols  ;  let  a  =  2,  b  =  3, 
c  =r  i,/=  £;  then  the  given  quantities  to  be  subtracted,  — 


become  — 

(+3  -2-i+i), 
which  reduces  to  — 

(4  -  24)  =  ij. 

Thus  the  quantity  to  be  substracted  equals  ij.  Applying 
the  numerals  to  the  above  expression  — 

A  —  b+  a  +f  —  c 
becomes  — 

A  —  3  +  2  +  -J  —  i  =A  —  4+2%  =  A  —  i-J. 

A  correct  result  ;  it  is  the  same  as  before.  Restoring  now 
the  symbols  represented  by  A>  we  have  for  the  whole  ex- 
pression — 


which,  by  cancelling  (Art.  403)  and  by  adding  like  symbols 
with  like  signs,  reduces  to  — 

2  a  —  2  c. 

To  test  this  result,  let  the  quantity  which  was  represented 
by  A  have  the  proper  numerals  substituted,  thus  : 

+  0  +  b  —  c  —  /, 

+  2  +  3  ~  i  -4=5-  i*  =  3i- 


4O2  ALGEBRA.    . 

The  sum  of  the  given  quantity  required  to  be  subtracted 
was  before  found  to  amount  to  i^,  therefore  — 

A  -1$ 
becomes  — 


And  the  result  by  the  symbols  as  above  was  — 

2  a  —  2  c, 
which  becomes  — 

2X2  —  2X1, 

or  — 

4  —  2  —  2; 

a  result  the  same  as  before,  proving  the  work  correct.  An 
examination  of  the  signs  in  the  above  expression,  which  de- 
notes the  problem  performed,  will  show  that  the  sign  of  each 
symbol  which  was  required  to  be  subtracted  has  been 
changed  in  the  operation  of  subtraction.  Before  subtract- 
ing they  were  — 


after  subtraction  they  are  — 

(-b  +  a+f-c). 

By  this  result  we  learn,  that  to  subtract  a  quantity  we  have 
but  to  change  its  sign  and  annex  it  to  the  quantity  from 
which  it  was  required  to  be  subtracted. 

Example  :  Subtract  a  —  b  from  c  +  d.    Answer,  c  +  d  —  a  +  b. 

If  numerals  be  substituted,  say  a  =  7,  b  =  4,  c  =  5,  and 
d=g,  then— 

c+d      becomes     5+9=14, 
a-b  "  ;_4=    3, 

c  +  d  —  (a  —  b)  —  14  —  3—  ii, 
So,  also,  — 

c  +  d  —  a  +  b 
becomes  — 

+     —    +     =  ii. 


FRACTIONS   ADDED  AND    SUBTRACTED.  403 

405.  —  Algebraic  Fraction*:   Added  and  Subtracted.  — 

When  algebraic  fractions  of  like  denominators  are  to  be 
added  or  subtracted,  the  same  rules  (Arts.  385  and  386)  are 
to  be  observed  as  in  the  addition  or  subtraction  of  numeri- 
cal fractions  —  namely,  add  or  subtract  the  numerators  for  a 
new  numerator,  and  place  beneath  the  sum  or  difference  the 
common  denominator. 

For  example,  what  is  the  sum  of  T>  T»  T? 

bob 

For  this  we  have  — 


Subtract  ->  from  -;.     For  this  we  have  — 
a  a 

b-c 

d    ' 

What  is  the  algebraical  sum  of  — 

ben  r^ 

-  -  -  -  and  -  5? 

For  these  we  have— 

b  +  c  —  n  —  r 


To  exemplify  this,  let  b  represent  9,  c  =  8,  n  =  2,  r  =  3, 
and  d—  12. 

Then,  for  the  algebraic  sum,  we  have — 

0  +  8  —  2  —  3        12 

—  =  —  =  i. 

12  12 

Now,  taking  the  positive   and   negative   fractions  sep- 
arately, we  have— 

£  ..  *-  =  !?. 

12  "h    12  12  ' 

and — 

n2  nJ^-I 

12        12  12   " 


404  ALGEBRA 

Together — 


12.  -5  _  _£^  = 

12      12     ~~     12   ~~      ' 


as  before. 


406. — The  Least  Common  Denominator. — When  the 
denominators  of  algebraic  fractions  differ  it  is  necessary  be- 
fore addition  or  subtraction  can  be  performed  to  harmonize 
them,  as  in  the  reduction  of  the  denominators  of  numerical 
fractions  (Arts.  388-390).  For  example,  add  together  the 

/7i       @      y 

fractions  7—,  -7,  — .      In   these    denominators   we    perceive 
oc     o  ac 

that  they  collectively  contain  the  letters  a,  b  and  c,  and  no 
others.  It  will  be  requisite,  therefore,  that  each  of  the  frac- 
tions be  modified  so  that  its  denominator  shall  have  these 
three  factors.  To  effect  this  it  will  be  seen  that  it  is  neces- 
sary to  multiply  each  fraction  by  that  one  of  these  letters 
which  is  lacking  in  its  denominator.  Thus,  in  the  first,  a  is 

lacking,  therefore  (Art.  380)  T—  —7—.     In  the  second  a 

and  c  are  lacking,  therefore  T  — r-,  and  in  the  third 

by,ac-=-abc 

r  x  b  —    rb  . 

b  is  lacking,  therefore   —      ,  _  ^-^>     Placing  them  now 

together  we  have — 

aa  +  ace±br         a       e       r 

t |     


a  b  c.  be      b      a  c 

The  factor  a  a  may  be  represented  thus  #2,  which  means 
that  a  occurs  twice,  the  small  figure  at  the  top  indicating 
the  number  of  times  the  letter  occurs ;  a 2  is  called  a  squared, 
a  a  a  =  a*,  and  is  called  a  cubed. 

In  order  to  show  that  the  above  fraction,  resulting  as  the 
sum  of  the  three  given  fractions,  is  correct,  let  a  =  2,  b  —  3, 
c  =  4,  e  =  5,  and  r  =  6.  Then  the  three  given  fractions 
are — 

2    JU    6    =:  1:+ 1  + 1 


3x4      3      2x4       6      3      4 


FRACTIONS   SUBTRACTED.  405 

In  equalizing  these  denominators  we  multiply  the  second 
fraction  by  2,  and  the  third  by  i£,  which  will  give  — 

5_  x  2  =  10^  3x^i_4i. 
3x2=   6  '  4  x  i£  ~  '    6  ' 
then— 

1  •  1°_  ,44       ij_4       03i          _7_ 
6+  6  "  6   '       6  6   "      12* 


Now  the  sum  of  the  fractions  i 


s  — 


22+2   X4X    5    +   3X6 
tJi , 

2X3X4 

4  +  40+  1 8  __  62  _        14  _         7 
24  24  "    2  24  ~    2  12  ' 

the  same  result  as  before,  thus  showing  that  the  reduction 
was  rightly  made. 

407. — Algebraic  Fractions  Subtracted. — To  exemplify 
the  subtraction  of  fractions,  let  it  be  required  to  find  the 

algebraic  sum  of  -  -  -%  —  j.  These  denominators  all  dif- 
fer. The  fractions,  therefore,  require  to  be  modified,  so 
that  each  denominator  shall  contain  them  all.  To  accom- 
plish this,  the  first  fraction  will  need  to  be  thus  treated  : 

ax df= adf 
7x 
the  second — 

_  b_xcf=  _  bcf 

~    dXCf=     :,.      Cdf'' 

the  third— 

e  x  c  d  =        c  d e 

~  fxcd=  ~  7d~f 
The  sum  of  these  is — 

adf—  bcf  —  c  de 

~Tdf~        ' 


406  ALGEBRA. 

That  this  is  a  correct  answer,  let  the  result  be  proved  by 
figures  ;  thus,  for  a  put  15  ;  b,  2  ;  c,  3  ;  d,  4;  e,  5  ;  /,  6.     Then' 
we  shall  have  — 

a       b        e        15         25 
~c~~d~J~-    T    "  4"  6* 

It  will  be  observed  that  these  denominators  may  be  equal- 
ized by  multiplying  the  first  fraction  by  2,  and  the  second 
by  ij,  therefore  we  have  — 

j$o       _3  _    5. 
6   "     6       6' 

To  make  the  required  subtraction  we  are  to  deduct  from  30 
(the  numerator  of  the  positive  fraction),  first  3,  then  5  ;  or, 
the  sum  of  the  numerators  of  the  negative  fractions  ;  or  for 
the  numerator  of  the  new  fraction  we  have  30  —  8  —  22. 
The  required  result,  therefore,  is  — 


-.~, 
63 

To  apply  this  test  to  the  algebraic  sum  we  have  — 

a  d  f  —  b  c  f  —  c  d  e        I5x4x6  +  2x  3x64.3x4x5 
"  ~  cdf  3x4x6 

which  by  multiplication  reduces  to— 

360  —  36  —  60  _    264  _  22_        ri.         a 
~W  '-  72"  :  :    6    :  :    3    : 

a  result  the  same  as  before,  proving  the  work  correct.     An 
other  example  : 

a        b  c    d        .    e 

From  -----  take  -,  —  and  -  ; 
n       m  n    m  n 

or.  find  the  algebraic  sum  of— 

a     .  b          c         d_        e_ 
n      m        n  ,     m        n 


DENOMINATORS   HARMONIZED.  407 

The  fractions  which   have  the  same   denominator  may  be 
grouped  together  thus : 


a        c        e        a  —  c  —  c 


n        n       n  n 

and — 

*       A       ^L       b~d 

m         m  m 

To  harmonize  these  two  denominators,  m  and  n,  the  first 
fraction  must  be  multiplied  by  m  and  the  last  by  ;/,  or — 

m  (a  —  c  —  e)        n  (b  —  d}        m  (a  —  c  —  e)  +  n  (b  —d) 
m n  m n  mn 

In  the  polynomial  factor  within  the  parentheses  (a  —  c  —  e)  we 
have  the  positive  quantity  a,  from  which  is  to  be  taken  the 
two  negatives  c  and  e,  or  their  sum  is  to  be  taken  from  #,  or 
(a  —  (c  +  e) ).  With  this  modification  we  have  for  the  alge- 
braic sum  of  the  five  given  fractions — 

m(a  —  (c  +c))  +  n  (b  —  d) 
mn 

To  test  the  accuracy  of  this  result,  let  the  value  of  the  sev- 
eral letters  respectively  be  as  follows :  a  =  n,  b  =g,  c  =  3, 
d  =  4,  e  =  5,  m  =  10,  and  n  —  8.  Then  the  sum  is — 

10  (11  -(3 +  5)) +  8  (9-4)  __  TO  __  7 
10  x  8  "80       8* 

Now,  taking  the  fractions  separately,  we  have— 

<*__£.       £_Ii       (A,l\       ii_8       1 
n       n        n  ~     8         V8  +  8/  "    8         8  "  8' 

^^945 
again-  —  -  -  —  =  =  —  -  —  ==  — ; 

or,  together  we  have,  as  the  sum  of  these  two  results— 

8  +  lo' 


408 


ALGEBRA. 


To  harmonize  these  denominators  we  may  multiply  the  first 
fraction  by  5,  and  the  second  by  4,  thus : 


8x5=4°'    io  x  4  =  4°' 
and  then  the  sum  is — 

Jl      ™  -  35.  =  1_. 
40  +  40  ""  40         8  ' 

the  same  result  as  before,  thus  the  accuracy  of  the  work  is 
established. 

408. — Graphical   Representation  of  multiplication. — 

In  Fig.  278,  let  A  BC  D,  a  rectangle,  have  its  sides  A  B  and 

A  B 


FIG.  278. 

A  C  divided  into  equal  parts.  Then  the  area  of  the  figure 
will  be  obtained  by  multiplying  one  side  by  the  other,  or 
putting  a  for  the  side  A  B,  and  b  for  the  side  A  C,  then  the 
area  will  be  a  x  b,  or  ab.  This  will  be  the  correct  area  of 
the  figure,  whatever  the  length  of  the  sides  may  be.  If,  as 
shown,  the  area  be  divided  into  4  x  7  =  28  equal  rectangles, 
then  a  would  equal  7,  and  b  equal  4,  and  a  b  =  7  x  4  =  28,  the 
area.  If  A  B  equal  28  and  A  C  equal  16,  then  will  a  —  28, 
and  b  =  16,  and  a  b  =  28  x  16  =  448,  the  area. 

409. — Graphical  Multiplication  :  Three  Factor§. — Let 

A  B  C D  E  FG  (Fig.  279)  represent  a  rectangular  solid  which 
may  be  supposed  divided  into  numerous  small  cubes  as 
shown.  Now,  if  a  be  put  for  the  edge  A  B,  b  for  the  edge 
A  C,  and  c  for  the  edge  CD,  then  the  cubical  solidity  of  the 


MULTIPLICATION   OF  A  BINOMIAL. 


409 


whole  figure  will  be  represented  by  a  x  b  x  c  =  a  b  c.  If  the 
edge  A  B  measures  6,  the  edge  A  C  3,  and  the  edge  CD  4, 
then  abc  =  6x3x4  =  72  =  the  cubic  contents  of  the 
figure,  or  the  number  of  small  cubes  contained  in  it. 


DL 


/G 


FIG.  279. 

410. — Graphic  Representation:  Two  and  Three  Fac- 
tors— Figs.  278  and  279  serve  to  illustrate  the  algebraic  ex- 
pressions a  b  and  a  b  c.  In  the  former  it  is  shown  that  the 
multiplication  of  two  lines  produces  a  rectangular  surface, 
or  that  if  a  and  b  represent  lines,  then  a  b  may  represent  a 
rectangular  surface  (Fig.  278)  having  sides  respectively 
equal  to  a  and  b.  And  so  if  a,  b,  and  c  represent  three  sev- 
eral lines,  then  a  b  c  may  represent  a  rectangular  solid 
279)  having  edges  respectively  equal  to  #,  £,  and  c. 

A  BE 


FIG.  280. 

4f|.  —  Graphical   Multiplication   of  a   Binomial.  —  Let 

A  B  CD  (Fig.  280)  be  a  rectangular  surface,  and  BED  F  an- 
other rectangular  surface,  adjoining  the  first.  The  area  of 
the  whole  figure  is  evidently  equal  to — 

(A  B  +  B  E)  x  A  C. 


410  ALGEBRA. 

The  area  is  also  equal  to — 


ABxA  C  +  BExBD: 
or,  since  A  C  —  B  D,  the  area  equals 


ABxAC  +  BExAC; 

or,  if  symbols  be  put  to  represent  the  lines;  say  a  for  A  B, 
b  for  B  E,  and  c  for  A  C,  then  the  two  representatives  of  the 
area,  as  above  shown,  become  :  The  first — 

(a  +  b}  x  c  =  area ; 
and  the  last — 

(a  x  c)  +  (b  x  c)  =  area. 
Hence  we  have — 

(a  +  b}  c  =  a  c  -\-  b  c. 

This  result  exemplifies  the  algebraic  multiplication  of  a  bi- 
nomial, which  is  performed  thus :  Let  a  +  b  be  multiplied 
by  c. 

The  problem  is  stated  thus  : 

(a  +  b)  c. 
To  perform  the  multiplication  indicated  we  Droceed  thus : 

a  +  b 
c 


ac  +  be 

multiplying  each  of  the  factors  of  the  multiplicand  sepa- 
rately and  annexing  them  by  the  sign  for  addition.  Putting 
the  two  together,  or  showing  the  problem  and  its  answer  in 
an  equation,  we  have — 

(a  +  b)  c  ==  a  c  +  b  c, 

producing  the  same  result,  above  shown,  as  derived  from 
the  graphic  representation. 

412.— Graphical  Squaring  of  a  Binomial.— Let  EGCJ 
(Fig:  281)  be  a  rectangle  of  equal  sides,  and  within  it  draw 


SQUARING  OF  A   BINOMIAL.  411 

the  two  lines,  A  H  and  F  D,  parallel  with  the  lines  of  the 
rectangle,  and  at  such  a  distance  from  them  that  the  sides, 
A  B  and  B  D,  of  the  rectangle,  A  B  C  D,  shall  be  of  equal 
length.  We  then  have  in  this  figure  the  three  squares, 
E  GCJ,  AB  CD,  and  FGBH,  also  the  two  equal  rect- 
angles, EFA  B  and  BHD  J. 

Let  E  F  be  represented  by  a  and  F  G  by  b,  then  the  area 
of  ABC  D  will  be  axa  =  a*  \  the  area  of  FGBH  will  be 
b  x  b  —  b 2 ;  the  area  of  E  F  A  B  will  be  a  x  b  =  a  b,  and  that 


A                      B 

• 

C                        1 

5 

FIG.  281. 

of  B  H  D  y  will  be  the  same.     Putting  these  areas  together 
thus— 


the  sum  equals  the  area  of  the  whole  figure  —  equals  the  prod- 
uct of  EG  x  E  £*—  -equals  the  product  — 

(a  +  b)  x(a  +  b). 
So,  therefore,  we  have  — 

(a  +  b)  (a  +  b}  =  a*  +  2a&  +  &';  (112.) 

or,  in  general,  the  square  of  a  binomial  equals  the  square  of 
the  first,  plus  twice  the  first  by  the  second,  plus  the  square  of  the 
second.  This  result  is  obtained  graphically.  The  same  re- 
sult may  be  obtained  by  algebraic  multiplication,  combining 


412  ALGEBRA. 

each  factor  of  the  multiplier  with  each  factor  of  the  multi- 
plicand and  adding  the  products,  thus — 

a  +  b 
a  +  b 


a  b 


The  same  result  as  above  shown  by  graphical  representa- 
tion. 

413.  —  Graphical  Squaring  of  the  Difference  of  Two 
Factors.  —  Let  the  line  E  C  (Fig.  281)  be  represented  by  c, 
and  the  line  A  E  and  A  C  as  before  respectively  by  b  and 
a,  then  — 


c  —  b  —  a. 
From  this,  squaring  both  sides,  we  have  — 


The  area  of  the  square  A  B  C  D  may  be  obtained  thus  : 
From  the  square  E  G  C  J  take  the  rectangle  E  G  x  E  A  and 
the  rectangle  F  G  x  D  y,  minus  the  square  F  G  B  H,  or 
from  c*  take  the  rectangle  cb,  and  the  rectangle  c  b,  minus 
the  square,  b  a,  and  the  remainder  will  be  the  square,  a  8  ;  or, 
in  proper  form  — 


In  deducting  from  c*  the  rectangle  cb  twice,  we  have  taken 
away  the  small  square  twice  ;  therefore,  to  correct  this 
error,  we  have  to  add  the  small  square,  or  b*.  Then,  when 
reduced,  the  expression  becomes  — 


This  result  is  obtained  graphically.     The  result  by  algebraic 


PRODUCT   OF  THE   SUM   AND   DIFFERENCE. 


413 


process  will  now  be  sought.     The  square  of  a  quantity  may 
be  obtained  by  multiplying  the  quantity  by  itself,  or — 


(c  -  £)2  =      - 


(US-) 


In  this  process,  as  before,  each  factor  of  the  multiplier  is 
combined  with  each  factor  of  the  multiplicand  and  the  sev- 
eral products  annexed  with  their  proper  signs  (Art.  415), 
and  thus,  by  algebraic  process,  a  result  is  obtained  precisely 
like  that  obtained  graphically.  This  result  is  the  square  of 
the  difference  of  c  and  b ;  and  since  c  and  b  may  represent 
any  quantities  whatever,  we  have  this  general — 

RULE. —  The  square  of  the  difference  of  two  quantities  is 
equal  to  the  sum  of  the  squares  of  the  two  quantities,  minus  twice 
their  product. 


FIG.  282. 


4(4. Graphical  Product  of  the  Sum  and  Difference 

of  Two  Quantities. — Let  the  rectangle  A  B  C  D  (Fig.  282) 
have  its  sides  each  equal  to  a.  Let  the  line  E  F  be  parallel 
with  A  B  and  at  the  distance  b  from  it,  also,  the  line  F  G 
made  parallel  with  B  D,  and  at  the  distance  b  from  it.  Then 
the  line  E  F  equals  a  -f  /;,  and  the  line  E  C  equals  a  —  b. 
Therefore  the  area  of  the  rectangle  E  F  C  G  equals  n  +  b, 


4H  ALGEBRA. 

multiplied  by  a  —  b.     From  the  figure,  for  the  area  of  this 
rectangle,  we  have — 

ABCD-ABEH+HFDG  =  RFC  G\ 
or,  by  substitution  of  the  symbols, 

a 2  —  a  b  +  b  (a  —  B). 
Multiply  the  last  quantity  thus— 

a-b 
b 


ab-b*  =  b(a-b}. 
Substituting  this  in  the  above  we  have  — 

a*  —  a  b  +  a  b  —  b*  =  (  a  T  &)  x  (a  —  b). 

Two  of  these  like  quantities,  having  contrary  signs,  cancel 
each  other  and  disappear,  reducing  the  expression  to  this— 


The  correctness  of  this  result  is  made  manifest  by  an  inspec- 
tion of  the  figure,  in  which  it  is  seen  that  the  rectangle  E  FC  G 
is  equal  to  the  square  A  BCD  minus  the  square  BJHF. 
For  ABEH  equals  BJDG.  Now,  if  from  the  square 
A  B  CD  we  take  away  A  B  E  H,  and  place  it  so  as  to  cover 
BJDG,  we  shall  have  the  rectangle  E  FC  G  plus  the  square 
BJHF-,  showing  that  the  square  A  BCD  is  equal  to  the 
rectangle  EFC  '  G  plus  the  square  B  J  H  F  ';  or  — 

a*=(a  +  b)  x  (a-b)  +  b\ 

The  last  quantity  may  be  transferred  to  the  first  member  of 
the  equation  by  changing  its  sign  (Art.  403).     Therefore  —  - 


+  b}  x  (a  -  b\ 
as  was  before  shown. 


MULTIPLICATION — PLUS  AND   MINUS. 


415 


The  result  here  obtained  is  derived  from  the  geometrical 
figure,  or  graphically.  *  Precisely  the  same  result  may  be 
obtained  algebraically  ;  thus — 


a  +  b 
a-  b 


a*  +ab 
-ab-b* 


(.114.) 


Here  the  two  like  quantities,  having  unlike  signs,  cancel 
each  other  and  disappear,  leaving  as  the  result  only  the  dif- 
ference of  the  squares. 

The  result  here  obtained  is  general ;  hence  we  have  this — 

RuL.E. —  The  product  of  the  sum  and  difference  of  two  quan- 
tities equals  the  difference  of  their  squares. 


^                              G          1 

E                           J 

F 

FIG.  283. 

415. — PIu§  and  Minus  Sign§  in  Multiplication. — In  pre- 
vious articles  the  signs  in  multiplication  have  been  given  to 
products  in  accordance  with  this  rule,  namely :  Like  signs 
give  plus  ;  unlike  signs,  minus.  This  rule  may  be  illustrated 
graphically,  thus :  In  the  rectangular  Fig.  283,  let  it  be  re- 
quired to  show  the  area  of  the  rectangle  A  G  C H,  in  terms 
of  the  several  parts  of  the  whole  figure.  Thus  the  area  of 
A  GE  J  equal  ABEF-GBJF*n&  the  area  of  EJCH 
equals  E  FCD  -  J  F  H  D.  And  the  areas  of  A  G  E  J -r 
EJCH  equals  the  area  of  A  G  C  H.  Therefore  the  sum  of 
the  two  former  expressions  equals  A  G  C  H.  Thus  A  B£F— 
GBJF+ EFCD  —  JFHD  =  AGCH.  Let  the  several 
lines  now  be  represented  by  algebraic  symbols ;  for  example, 


416  ALGEBRA. 


let  AB  =  EF=a;  let  GB  =  ?F=  l>;  let  A  E  =  G  J  =  c  ; 
and  EC  —  J  H  =  d,  and  let  these  symbols  be  substituted  for 
the  lines  they  represent,  thus  ABEF-  GBJF+EFCD  - 
JFHD  =  AGCH. 

ac  —  be  +  ad  —  b  d  =  (a  —  b)  x  (c  +  d). 

An  inspection  of  the  figure  shows  this  to  be  a  correct 
result.  It  will  now  be  shown  that  an  algebraical  multiplica- 
tion of.  the  two  binomials,  allotting  the  signs  in  accordance 
with  the  rule  given,  will  produce  a  like  result.  For  example  — 

a-b 

c  +  d 


ac  —  be  +  ad  —  b  d. 


416. — Equality  of  Squares  on  Hj  potliemise  and  Sides  of 
Right-Angled  Triangle. — The  truth  of  this  proposition  has 
been  proved  geometrically  in  Art.  353.  It  will  now  be 
shown  graphically  and  proved  algebraically. 

Let  A  BCD  (Fig.  284)  be  a  rectangle  of  equal  sides,  and 
BED  the  right-angled  triangle,  the  squares  upon  the  sides 
of  which,  it  is  proposed  to  consider.  Extend  the  side  BE 
to  F;  parallel  with  BF  draw  DG,  C K,  and  A  L.  Parallel 
with  ED  draw  A  J  and  L  G.  These  lines  produce  triangles, 
AHB,  AC?,  ALC,  CKD,  and  C  G  D,  each  equal  to  the 
given  triangle  BED  (Art.  337).  Now,  if  from  the  square 


SQUARES  ON  RIGHT-ANGLED  TRIANGLE.  417 

A  B  CD  we  take  Afiffand  place  it  at  CD  G  ;  and  if  we  take 
BED  and  place  it  at  A  L  C  we  will  modify  the  square 
A  BCD,  so  as  to  produce  the  figure  LGDEHAL,  which 
is  made  up  of  two  squares,  namely,  the  square  D  E  FG  and 
the  square  ALFH,  and  these  two  squares  are  evidently 
equal  to  the  square  A  B  CD.  Now,  the  square  D  E  FG  is  the 
square  upon  ED,  the  base  of  the  given  right-angled  triangle, 
and  the  square  A  L  F  H  is  the  square  upon  A  H  =  BE,  the 
perpendicular  of  the  given  right-angled  triangle,  while  the 
square  A  B  C  D  is  the  square  upon  B  D,  the  hypothenuse  of 
the  given  right-angled  triangle.  Thus,  graphically,  it  is  shown 
that  the  square  upon  the  hypothenuse  of  a  right-angled  triangle 
is  equal  to  the  sum  of  the  squares  upon  the  remaining  two  sides. 
To  show  this  algebraically,  let  B  E,  the  perpendicular  of 
the  given  right-angled  triangle,  be  represented  by  a  ;  E  D, 
the  base,  by  b,  and  B  D,  the  hypothenuse,  by  c.  Then  it  is 
required  to  show  that— 


Now,  since  D  K  ==  B  E  =  a,  therefore,   E  K  =  E  D  - 
D  K  =  b  —  a,  and  the  square  E  K  J  H  equals  (b  —  #)2,  which 
(Art.  413)  equals 


This  is  the  value  of  the  square  EKJ  H  which,  with  the  four 
triangles  surrounding  it,  make  up  the  area  of  the  square 
A  B  C  D.  Placing  the  triangle  A  B  H  of  this  square  outside 
of  it  at  CD  G,  and  the  triangle  B  E  D  at  A  L  C,  we  have  the 
four  triangles,  grouped  two  and  two,  and  thus  forming  the 
two  rectangles  C  G  D  K  and  A  L  C  J.  Each  of  these  rect- 
angles has  its  shorter  side  (A  L,  C  G)  equal  to  BE  —  a,  and 
its  longer  side  L  C,  G  D,  equal  to  E  D  =  b  ;  and  the  sum  of 
the  two  rectangles  is  ab  +  ab=2ab.  This  represents  the 
area  of  the  two  rectangles,  which  are  equal  to  the  four  tri- 
angles, which,  together  with  the  square  EKJH,  equal  the 
square  ABCD\  or— 

ABCD^EKJH+CGDK+ALCJ, 


ALGEBRA. 

or —  c*  —  (b  —  a)*  +  a  b  +  a  b,  or— 

c*=  (b  -  tf)2  +  2ab. 

Then,  substituting  for  (b  —  #)2,  its  equivalent  as  above,  we 
have — 

c*  =  b*  —  2ab  +  a*+  2ab. 

Remove  the  two  like  quantities  with  unlike  signs  (Art.  402), 
and  we  have — 

c*=b*+a*-,  (115.) 

which  was  to  be  proved. 

417.—  Division  the  Reverse  of  Multiplication. — As  di- 
vision is  the  reverse  of  multiplication,  so  to  divide  one  quan- 
tity by  another  is  but  to  retrace  the  steps  taken  in  multipli- 
cation. If  we  have  the  area  ab  (Fig.  278),  and  one  of  the 
factors  a  given  to  find  the  other,  we  have  but  to  remove 
from  a  b  the  factor  a,  and  write  the  answer  b. 

If  we  have  the  cubic  contents  of  a  solid  abc  (Fig.  279), 
and  one  of  the  factors  a  given  to  find  the  area  represented 
by  the  other  two,  we  have  but  to  remove  a,  and  write  the 
others,  b  c,  as  the  answer. 

If  there  be  given  the  area  represented  by  a  (b  +  c)  (see 
Art.  41 1),  and  one  of  the  factors  a  to  find  the  other,  we  have 
but  to  remove  a  and  write  the  answer  b  +  c.  Sometimes,  how- 
ever, a  (b  +  c)  is  written  ab  +  ac.  Then  the  given  factor  is 
to  be  removed  from  each  monomial  and  the  answer  written 
b  +  c. 

If  there  be  given  the  area  represented  by  a*  +  2  ab  +  b* 
to  find  the  factors,  then  we  know  by  Art.  412  that  this  area 
is  that  of  a  square  the  sides  of  which  measure  a+  b,  and  that 
the  area  is  the  product  of  a  +  b  by  a  +  b ;  or,  that  a  +  b  is 
the  square  root  of  a1  +  2  a  b  +  b*. 

If  there  be  given  the  area  a~  •-  2  ab  +  b~  to  find  its  fac- 
tors, then  we  know  by  Art.  413  that  this  area  is  that  of  a 
square  whose  sides  measure  a  —  b,  or  that  it  is  the  product 
of  a  —  b  bv  a  —  b,  or  the  square  of  a  —  b' 


PROCESSES   IN   DIVISION.  419 

If  there  be  given  the  difference,  of  the  squares  of  two 
quantities,  or  the  area  represented  by  a*  —  b*y  to  find  its  fac- 
tors, then  we  know  by  Art.  414  that  this  is  the  area  pro- 
duced by  the  multiplication  of  a  —  b  by  a  +  b. 

4-18. — BH  vi  si  on  :  Statement  of  Quotient. — In  any  case  of 
division  the  requirement  may  be  represented  as  a  fraction ; 
thus  :  To  divide  c  +  d  —  /by  a  —  ^  we  write  the  quotient 
thus— 

c  +  d-f 
a-  b 

For  example,  to  illustrate  by  numerals,  let  a  =  7,  b  =  3, 
c  =  4,  d  =  5,  and/  =  6.     Then  the  above  becomes — 


7-3         "4* 

4 1 9. — Division  ;  Reduction. — When  each  monomial  in 
either  the  numerator  or  denominator  contains  a  common 
quantity,  that  quantity  may  be  removed  and  placed  outside 
of  parentheses  containing  the  monomials  from  which  it  was 
taken  ;  thus,  in — 

2  ab  -\.  ^  ac  —  8  ad 

~T~ 

we  have  2  and  a  factors  common  to  each  monomial  of  the 
numerator.     Therefore  the  expression  may  be  reduced  to 

2  a  (b  +  2  c  —  4^) 


To  test  this  arithmetically  we  willl  put  a  =  9,  b  =  7,  c  —  5, 
d  =  4,  and  /  =  6.     Then  for  the  first  expression  we  have— 

2x9x7  +  4x9x5  —  8x9x4 
~6~~ 


which  equals — 

126  +  1 80  —  288 


42O  ALGEBRA. 

And  for  the  second  expression — 

2  x  9  (7  +  2  x  5  —  4  x  4) 

6 
which  equals — 

18  (17  &  1 6)        18 


the  same  result  as  before.  It  will  be  observed  that  in  this 
process  of  removing  all  common  factors  algebra  furnishes 
the  means  of  performing  the  work  arithmetically  with  many 
less  figures.  The  reduction  is  greater  when  the  common 
factors  are  found  in  both  numerator  and  denominator.  For 
example,  in  the  expression  — 

$  an  +  9  fin  —  15  en 

12  dn  — 


we  have  3  n  a  factor  common  to  each  monomial  in  the  nu- 
merator and  denominator  ;  therefore  the  expression  reduces 
to 


And  now,  since  3  n  is  a  factor  common  to  both  numerator 
and  denominator,  these  cancel  each  other  ;  therefore  (Art. 
371)  the  expression  reduces  to— 


—  5 


To  test  these  reductions  arithmetically,  let  a  =  9,  b  =  8, 
c  —  4,  d  —  6,  /=  3,  and  n  =  .5.  Then  the  first  expression 
becomes  — 


3x9x5  _+_9_  x  8  x  5  -  15x4x5 
12x6x5  —  18x3x5 

which  equals — 

135  +  360-300^     *95  _     i_. 
360  -  270  90          6  ' 


FORMULA   OF  THE   LEVER.  421 

and  the  second  expression  becomes — 


9  +  3x8  —  5x4 
4x6  —  6x3   ' 

which  equals— 

9  +  24  —  20  __  ^S  __    1 
24—18  6          6* 

The  same  result,  but  with  many  less  figures. 

420.  —  Proportionals  :  Analysis.  —  In  the  formula  of  the 
lever  (Art.  377),  P  x  CF  =  R  x  E  C.  Let  n  be  put  for  the 
arm  of  leverage  6^and  m  for  E  C.  Then  we  have— 

Pn  =  Rm, 
from  which  by  division  (Art.  372)  we  have  (Art.  399)— 


and~ 


(in.) 


Suppose  there  be  a  case  in  which  neither  R  nor  P  severally 
are  known,  but  that  their  sum  is  known  ;  and  it  is  required 
from  this  and  the  m  and  n  to  find  R  and  P.  Let— 

W  =  R  +  P, 

then—  W-  R  =  P.  (See  Art.  403.) 

The  value  of  P  was  above  found  to  be— 


Since  P  =  R        and  also  equals    W  -  -  R,  therefore- 


422  ALGEBRA. 

Transferring  R  to  the  opposite  member  (Art.  403)  we  hav< 


Here  R  appears  as  a  common  factor  and  may  be  separated 
by  division  (Art.  419) ;  thus— 

W=  I 

By  division  the  factor  ( i  +  —  J  may  be  transferred  to  the 
opposite  member  (Art.  371).     Thus  we  have — 

W 


by  which  we  find  the  value  of  R  developed.     As  an  example, 
let  W  =  looo  pounds,  m  =  3  feet  and  n  —  7  feet  ;  then— 

1000  _  looo 
~-  i+l  =    "ToT  ' 

Multiplying  the  numerator  and  denominator  by  7,  we  get— 

7  x  1000 


=700. 


Since  —  R  +  P  —  1000, 

and—  R          =    700, 

then—  P          —   300. 

But  a  process  similar  to  the  above  develops  an  expression 
for  the  value  of  P,  which  is  — 


i  +  n 

~Z  ("70 

Putting  this  to  the  test  of  figures,  we  have  — 
looo        looo       3000 

r>  _      _  —   __  —  if  _  _  —   •inn 

•  i  +  i  -     y         10    -  3°°- 


NEGATIVE  EXPONENTS.  423 

4-21. —  Raising  a  Quantity  to  any  Power.  —  When  a 
quantity  is  required  to  be  multiplied  by  its  equal,  the  prod- 
uct is  called  the  square  of  the  quantity.  Thus  a  x  a  =  a* 
(Art.  412).  If  the  square  be  multiplied  by  the  original 
quantity  the  result  is  a  cube  ;  or,  a9  x  a  =  a3 ;  or,  generally, 
for-— 


a,  a  a,  a  a  a,  aaaa,  aaaaa, 
we  put— 


in  which  the  small  number  at  the  upper  right-hand  cor- 
ner indicates  the  number  of  times  the  quantity  occurs  in 
the  expression.  Thus,  if  a  —  2,  then  #a  =  2  x  2  =  4, 
a3  =  4  x  2  =  8,  a*  =  8  x  2  =  16,  a"  —  16  x  2  =  32  ;  any  term 
in  the  series  of  powers  may  be  found  by  multiplying  the 
preceding  one  by  a,  or  by  dividing  the  succeeding  one  by  a. 

Thus  a*  x  a  =  a*,  and  -  —  a\ 
a 


4-22. — Quantities  with  Negative  Exponents. — The  series 
of  powers,  by  division,  may  be  extended  backward.     Thus, 

a6  a*        3     a*        *    a*        ,     a1        0    a° 

if  we  divide  —  =  #  ;  —  =  a  ;   -    =  a  ;  —=  a  ',  -    =  a  ;  -    -a    ; 
a  #  #  a  a  a 

f?lW°;  "-=«>,  etc. 
a  a 

In  this  series  we  have  -  =  a*.     But  a  quantity  divided  by 

its  equal  gives  unity  for  quotient,  or  -  =  i.    Therefore,  —  =  i, 

and  a°  —  i.     This  result  is  remarkable,  and  holds  good  re- 
gardless of  the  value  of  a. 

From  this  and  the  preceding  negative  exponents  we  de- 
rive the  following : 


424  ALGEBRA. 


. 

a  =  —  =   -, 

a          a 


3      a~"         I          I 

a  -3=-  —  —  =  -,  etc. 


Showing  that  #  quantity  with  a  negative  exponent  may  have 
substituted  for  it  the  same  quantity  with  a  positive  exponent,  but 
used  as  a  denominator  to  a  fraction  having  unity  for  the 
numerator. 

423.  —  Addition  and  Subtraction  of  Exponential  Quan- 
tities. —  Equal  quantities  raised  to  the  same  power  may  be 
added  or  subtracted;  as,  «2  +  2«2=  3#2;  but  expressions  in 
which  the  powers  differ  cannot  be  reduced  ;  thus,  a*  +  a  —  a* 
cannot  be  condensed. 

4-24-.  —  multiplication   of  Exponential  Quantities.  —  It 

will  be  observed  in  Art.  421  that  in  the  series  of  powers,  the 
index  or  exponent  increases  by  unity  ;  thus,  a1,  a\  a\  a\  etc.  ; 
and  that  this  increase  is  effected  by  multiplying  by  the  root, 
or  original  quantity.     From  this   we  learn  that  to  multiply 
two  quantities  having  equal  roots  we  simply  add  their  exponents. 
Thus  the  product  of  a,  a'\  and  a3  is  a'  x  a*  x  a*  —  a\ 
The  product  of  a~2,  #3,  and  ab  is  a~*  x  a*  x  a*  =  a*. 
The  exponents  here,  are  :  —  2  +  3  +  5  —  8  —  2  =  6. 

425.  —  Division  of  Exponential  Quantities.  —  As  division 
is  the  reverse  of  multiplication,  to  divide  equal  quantities 
raised  to  various  powers,  we  need  simply  to  subtract  the  expo- 
nent of  the  divisor  from  that  of  the  dividend.  Thus,  to  divide 
a"  by  a9  we  have  a*"*  =  a\  That  this  is  correct  is  manifest  ; 
for  the  two  factors,  a*  x  a\  in  their  product,  a\  produce  the 
dividend. 

To  divide  a  by  a*,  we  have  a^  =  a~\  which  is  equal  to  -3 


EXPLANATION   OF   LOGARITHMS.  425 

(see  Art.  422).  The  same  result  may  be  had  by  stating  the 
question  in  the  usual  form.  Thus,  to  divide  a1  by  a"  we  have 

— 6,  a  fraction  which  is  not  in  the  lowest  terms,  for  it  may  be 

put  thus,  -T— 5  =  — ,  by  which  it  is  seen  that  it  has  in  both  its 
a  a        a 

numerator  and  denominator  the  quantity  a\  which  cancel 
each  other  (Art.  371).  Therefore,  -6  =  L ;  the  same  result 
as  before. 

426.— Extraction  of  Radicals.— We  have  seen  that  the 
square  of  a  is  a1  x  a1  =  a* ;  of  2  a3  is  2  a*  x  2  a3  =  4  a6 ;  in  each 
case  the  square  is  obtained  by  doubling  the  exponent. 

To  obtain  the  square  root  the  converse  follows,  namely, 
take  half  of  the  exponent. 

Thus  the  square  root  of  a*  is  a1,  of  a?  is  a,  of  a6  is  a3. 

The  same  rule,  when  the  exponent  is  an  odd  number, 
gives  a  fractional  exponent,  thus  :  the  square  root  of  a3  is  cfr  ; 
or,  of  ab,  is  a*.  So,  also,  the  square  root  of  a,  or  a1,  is  a*. 
Therefore,  we  have  <?*  =  Va,  equals  the  square  root  of  a,  and 
the  cube  root  of  a1  =  cfr  =  Va. 

427. — Logarithms. — We  have  seen  in  the  last  article  the 
nature  of  fractional  exponents.  Thus  the  square  root  of  a" 
equals  eft,  which  may  be  put  a**.  In  this  way  we  may  have 
an  exponent  of  any  fraction  whatever,  as  #**.  Between  the 
exponents  2  and  3,  we  may  have  any  number  of  fractional 
exponents  all  less  than  3  and  more  than  2.  So,  also,  the 
same  between  3  and  4,  or  any  other  two  consecutive  num- 
bers. 

The  consideration  of  fractional  exponents  or  indices  has 
led  to  the  making  of  a  series  of  decimal  numbers  called 
logarithms,  which  are  treated  in  the  manner  in  which  expo- 
nents are  treated  ;  namely — 

To  multiply  numbers  add  their  logarithms. 

To  divide  numbers,  subtract  the  logarithm  of  the  divisor  from 
the  logarithm  of  the  dividend. 


426  ALGEBRA. 

To  raise  any  number  to  a  given  power,  multiply  its  logarithm 
by  the  exponent  of  that  po^ver. 

To  obtain  the  root  of  any  power,  divide  the  logarithm  of  the 
given  number  by  the  exponent  of  the  given  power. 

As  an  example  by  which  to  exemplify  the  use  of  loga- 
rithms :  What  is  the  product  of  25  by  375  ? 

We  first  make  this  statement : 

Log.  of    25-  =  i- 
"     375*  =2. 

Putting  at  the  left  of  the  decimal  point  the  integer  char- 
acteristic,  or  whole  number  of  the  logarithm  at  one  less  than 
the  number  of  figures  in  the  given  number  at  the  left  of  its 
decimal  point. 

To  find  the  decimal  part  of  the  required  logarithm  we 
seek  in  a  book  of  Logarithms  (such  as  that  of  Law's,  in 
Weale's  Series,  London)  in  the  column  of  numbers  for  the 
given  number  25,  or  250  (which  is  the  same  as  to  the  man- 
tissa) and  opposite  to  this  and  in  the  next  column  we  find 
7940  and  a  .place  for  two  other  figures,  which  a  few  lines 
above  are  seen  to  be  39 ;  annex  these  and  the  whole  number 
is  0-397940.  These  we  place  as  below  : 

Log.  of  25  •  =  i  •  397940. 

Now,  to  find  the  logarithm  of  375,  the  other  factor,  we 
turn  to  375  in  the  column  ol  numbers  and  find  the  figures 
opposite  to  it,  4031,  which  are  to  be  preceded  by  57,  the  two 
figures  found  a  few  lines  above,  making  the  whole,  -574031, 
which  are  placed  as  below,  and  added  together. 

Log.  of   25-  =  1-397940 

11    375-  =  2-574031 

The  sum  =  3-971971 

This  sum  is  the  logarithm  of  the  product.  To  find  the 
product,  we  seek  in  the  column  of  logarithms,  headed  o-, 
for  -971971,  the  decimal  part.  We  find  first  97,  the  first  two 


EXAMPLES   OF   LOGARITHMS.  427 

figures,  and  a  little  below  seeking  for  1971,  the  remaining 
four  figures,  we  find  1740,  those  which  are  the  next  less, 
and  opposite  these,  to  the  left,  we  find  7,  and  above  93,  or 
together,  937  ;  these  are  the  first  three  figures  of  the  required 
product. 

For  the  fourth  figure  we  seek  in  the  horizontal  column 
opposite  7  and  1740  for  1971,  the  remaining  four  figures  of 
the  logarithm,  and  find  them  in  the  column  headed  5. 

This  figure  5  is  the  fourth  of  the  product  and  completes 
it,  as  there  are  only  four  figures  required  when  the  integer 
number  of  the  logarithm  is  3.  The  completed  statement 
therefore  is — 

Log.  of     25-'  =  1-397940, 
"      "    375-  =  2-574031, 


9375  =  3-97I97I- 

Another  example  in  the  use  of  logarithms.  What  is  the 
product  of  3957  by  94360? 

The  preliminary  statement,  as  explained  in  last  article,  is — 

Log-     3957  =  3- 
"      94360  =  4- 

In  the  book  of  logarithms  seek  in  the  column  of  numbers 
for  3957.  In  the  first  column  we  find  only  395,  and  opposite 
to  this,  in  the  next  column,  we  find  a  blank  for  two  figures, 
above  which  are  found  59.  Take  these  two  figures  as  the 
first  two  of  the  mantissa,  or  decimal  part  of  the  required 
logarithm,  thus,  0-59.  Again,  opposite  395  and  in  the  col- 
umn headed  by  7  (the  fourth  figure  of  the  given  number), 
we  have  the  four  figures  7366.  These  are  to  be  annexed  to 
(o-  59)  the  first  two  obtained.  The  decimal  part  of  the  loga- 
rithm, therefore,  is  0-597366. 

To  obtain  the  logarithm  for  94360,  the  other  given  num- 
ber, we  proceed  in  a  similar  manner,  and,  opposite  943,  we 
find  0-97;  then,  opposite  943  and  in  column  headed  6,  we 
find  4788,  or,  together,  the  logarithm  is  0-974788.  The 
whole  is  now  stated  thus — 


428  ALGEBRA. 

Log.  of  3957  =  3-597366 

94360=  4-974788 

"      "    373382000  =  8-572154  =  sum  of  logs. 

The  two  logarithms  are  here  added  together,  and  their  sum 
is  the  logarithm  of  the  product  of  the  two  given  factors. 
The  number  corresponding  to  the  above  resultant  logarithm 
may  be  found  thus:  Look  in  the  column  headed  o  for  57, 
the  first  two  numbers  of  the  mantissa,  then  in  the  same 
column,  farther  down,  seek  2154,  the  other  four  figures  of 
the  mantissa;  or,  the  four  (1709)  which  are  the  next  less 
than  the  four  sought,  and  opposite  these  to  the  left,  in  the 
column  of  numbers,  will  be  found  373,  the  first  three  figures 
of  the  product ;  opposite  these,  to  the  right,  seek  the  four 
figures  next  less  than  2154,  the  other  four  figures  of  the  man- 
tissa. These  are  found  in  the  column  headed  3  and  are 
2058.  The  3  at  the  head  of  the  column  is  the  fourth  figure 
in  the  product.  From  2154,  the  last  four  figures  of  the  man- 
tissa, deduct  the  above  2058,  or — 

2154, 
2058, 

Remainder,  96. 

At  the  bottom  of  the  page,  opposite  the  next  less  number 
(3727)  to  that  contained  in  3733,  the  answer  already  found, 
seek  the  number  next  less  to  the  above  remainder,  96.  This 
is  92-8,  and  is  in  the  column  headed  8.  Then  8  is  the  next 
number  in  the  product.  From  96  deduct  92-8,  and  multi- 
ply it  by  10,  or — 

96 

92-8 

3-2  x  10=  32. 

Then,  in  the  same  horizontal  column,  seek  for  32  or  its  next 
less  number.  This  is  23-2,  found  in  column  2.  This  2  is  the 
next  figure  in  the  product.  Additional  figures  may  be  ob- 
tained by  the  table  of  proportional  parts,  but  they  cannot  be 


THE   SQUARE   OF  A   BINOMIAL.  429 

depended  upon  for  accuracy  beyond  two  or  three  figures. 
We  therefore  arrest  the  process  here. 

The  product  requires  one  more  figure  than  the  integer  of 
the  logarithm  indicates;  as  the  integer  is  8,  there  must  be 
nine  figures  in  the  product.  We  have  already  six  ;  to  make 
the  requisite  number  nine  we  annex  three  ciphers,  giving 
the  completed  product  — 

3957  x  9436o  =  373382000. 

By  actual  multiplication  we  find  that  the  true  product  in  the 
last  article  is  373382520.  In  a  book  of  logarithms,  carried  to 
seven  places,  the  required  result  is  found  to  be  373382500^ 
which  is  more  nearly  exact. 

The  utility  of  logarithms  is  more  apparent  when  there 
are  more  than  two  factors  to  be  multiplied,  as,  in  that  case, 
the  operation  is  performed  all  in  one  statement.  Thus: 
What  is  the  product  of  3-75,  432-95,  1712,  and  0-0327  ? 

The  statement  is  as  follows  : 

Log.    3-75  =  0-574031 

432-95  =  2-636438 

1712-  =  3-233504 

•0327  =  8- 


Product  =  90891.  =  4-958521 

16 


Explanations  of  working  are  given  more  in  detail  in  most  of 
the  books  of  logarithms. 

428. — Completing    the   Square   of   a    Binomial. — We 

have  seen  in  Art.  412  that  the  square  of  a  binomial  (a  +  &) 
equals  a"  +  2  ab  +  b* — a  trinomial — the  first  and  last  terms 
of  which  are  each  the  square  of  one  of  the  two  quantities, 
while  the  second  term  contains  the  second  quantity  multi- 
plied by  twice  the  first  quantity — 

In  analytical  investigations  it  frequently  occurs  that  an 
expression  will  be  obtained  which  may  be  reduced  to  this 
form  : 


43°  ALGEBRA. 

a*  +  mab=f,  (nS.) 

in  which  m  is  the  coefficient  of  the  second  term,  and  a  and  b 
are  two  quantities  represented  by  a  and  b  or  any  other  two 
symbols. 

A  comparison  of  this  expression  with  the  square  of  a  bi- 
nomial (112.)  contained  in  Art.  412,  shows  that  the  member 
at  the  left  comprises  two  out  of  the  three  terms  of  the  square 
of  a  binomial  ;  as  thus  — 

a*  +  2  a  b  +  b*, 

but  with  a  coefficient  m  instead  of  2.  It  is  desirable,  as  will 
be  seen,  to  ascertain  a  proper  third  term  for  the  given  ex- 
pression ;  or,  as  it  is  termed,  "  to  complete  the  square."  The 
method  by  which  this  is  done  will  now  be  shown. 

A  consideration  of  the  above  trinomial  shows  that  the 
third  term  is  equal  to  the  square  of  the  quotient  obtained  by 
dividing  the  second  term  by  twice  the  square  root  of  the 
first  ;  or  — 


Now  a  third  term  to  the  above  binomial,  equation  (i  18.),  may 
be  obtained  by  this  same  rule.     For.  example  — 


The  rule  for  the  third  term  then  is:  Divide  the  second  term 
by  twice  the  square  root  of  the  first,  and  square  the  quotient. 

As  an  example,  let  it;  be  required  to  find  the  third  term 
required  to  complete  the  square  in  the  expression — 

6  n  x  -f-  4**  =/, 

in  which  n  and  /  are  known  quantities  and  x  unknown. 
Putting  it  in  this  form— 

4_r2  +  6  nx  = /, 
and  dividing  by  4,  we  have — 


BINOMIALS  CONTINUED.  431 


4  4 

which  reduces  to — 


Now  applying  the  above  rule  for  finding  the  third  term,  we 
have — 


which  is  the  required  third  term.  To  complete  the  square 
we  add  this  third  term  to  both  members  of  the  above  re- 
duced expression,  and  have  — 


The  member  of  this  expression  at  the  left  is  the  completed 
square  of  a  binomial,  the  two  quantities  constituting  which 
are  the  square  roots  of  the  first  and  third  terms  respectively  ; 
or  x  and  f  n,  and  we  therefore  have  — 


-2n*+\-H,- 


and  now  taking  the  square  root  of  both  sides  of  the  expres- 
sion, we  have — 


and,  by  transferring  the  second  quantity  to  the  right  mem- 
ber, we  have — 


an  expression  in  which  x,  the  unknown  quantity,  is  made  to 
stand  alone  and  equal  to  known  quantities. 

The  process  of  completing  the  square  is  useful,  as  has 
been  shown,  in  developing  the  value  of  an  unknown  quan- 


432  ALGEBRA. 

tity  where  it  enters  into  an  expression  in  two  forms,  one  as 
the  square  of  the  other. 

As  an  example  to  test  the  above  result,  let/  —  256  and 
n  =  8.  Then  we  have  by  the  last  expression  for  the  value 
of*— 


__ 

4 


=   1/64+36  -  6, 


=  y  ioo  —  6, 

x  =  10  —  6  =  4. 

Now  this  value  of  x  may  be  tested  in  the  original  expres- 
sion — 

6  nx  +    4**  =  /, 

for  which  we  have  — 

6x8x4  +  4x42  =  /, 
192  +  64  =  /, 


the  correct  value  as  above. 

PROGRESSION. 

429.  —  Arithmetical  Progre§§ion.  —  In  a  series  of  num- 
bers, as  i,  3,  5,  7,  9,  etc.,  proceeding  in  regular  order,  in- 
creasing by  a  common  difference,  the  series  is  called  an 
arithmetical  progression  ;  the  quantity  by  which  one  num- 
ber is  increased  beyond  the  preceding  one  is  termed  the 
difference.  If  d  represent  the  difference  and  a  the  first  term, 
then  the  progression  may  be  stated  thus  — 

Terms—      i,     2,  3,  4,  5, 

a,      a  +  d,      a  +  2  d,      a  +  3  d,      a  +  4  d,  etc. 

The  coefficient  of  d  is  equal  to  the  number  of  terms  preced- 
ing the  one  in  which  it  occupies  a  place.  Thus  the  fifth 
term  is  a  +  ^d,  in  which  the  coefficient  4  equals  the  number 
of  the  preceding  terms. 

From  this  we  learn  the  rule  by  which  at  once  to  desig- 


ARITHMETICAL  PROGRESSION.  433 

nate  any  term  without  finding  all  the  preceding  terms.  For 
the  one  hundredth  term  we  should  have  a  +  99  d,  or,  if  the 
number  of  terms  be  represented  by  n>  then  the  last  term 
would  be  represented  by— 

/  =  a  +  (n  —  i)  d.  (H9-) 

For  example,  in  a  progression  where  a,  the  first  term,  equals 
i,  d  the  difference,  2,  and  n,  the  number  of  terms,  90,  the  last 
term  will  be— 

/  =  a  +  (n  --  i)  d  =  i  +  (.90  —  i)  2  =  179. 

Therefore,  to  find  the  last  term : 

To  the  first  term  add  the  product  of  the  common  difference 
into  the  number  of  terms  less  one. 

By  a  transposition  of  the  terms  in  the  above  expression, 
so  as  to  give  it  this  form— 

a  =  /  -  (n  -  i)</,  (120.) 

we  have  a  rule  by  which  to  find  the  first  term,  which,  in 
words,  is — 

Multiply  the  number  of  terms  less  one  by  the  common  differ- 
ence,  and  deduct  the  product  from  the  last  term  ;  the  remainder 
will  be  the  first  term. 

By  a  transposition  of  the  terms  of  the  former  expression 
to  this  form — 

/  -  a  =  (  n  -  i)  d, 

and  dividing  both  members  by  (n  -  i),  we  have — 

rf=^-=^.  (121.) 

n  -  i  ' 

which  is  a  rule  for  the  common  difference,  and  which,  in 
words,  is — 

Subtract  the  first  term  from  the  last,  and  divide  the  remain- 
der by  the  number  of  terms  less  one ;  the  quotient  will  be  the  com- 
mon difference. 

Multiplying  both  members  of  the  equation  (121.)  by 
(n  —  i)  and  dividing  by  d,  we  obtain— 


434  ALGEBRA. 

l-a 
n-1  =  -J-' 

Transferring  i  to  the  second  member,  we  have — 

l-a 
n  =       ,     +  i  ;  (122.) 

which  is  a  rule  for  finding  the  number  of  terms,  and  which, 
in  words,  is — 

Divide  the  difference  between  the  first  and  last  terms  by  the 
common  difference ;  to  the  quotient  add  unity,  and  the  sum  will 
be  the  number  of  terms. 

Thus  it  has  been  shown,  in  equations  (119,)  (120),  (121), 
and  (122),  that  when,  of  the  four  quantities  in  arithmetical 
progression,  any  three  are  given,  the  fourth  may  be  found. 

The  sum  of  the  terms  of  an  arithmetical  progression  may 
be  ascertained  by  adding  them ;  but  it  may  also  be  had  by  a 
shorter  process.  If  the  terms  are  written  in  order  in  a  hori- 
zontal line,  and  then  repeated  in  another  horizontal  line  be- 
neath the  first,  but  in  reversed  order,  as  follows : 

i,    3,    5,    7,    9,  11,  13,  15, 

15,  13,  11,    9,    7,    5,    3,     i, 

16,  16,  16,  16,  16,  16,  16,  16, 

and  the  vertical  columns  added,  the  sums  will  be  equal.  In 
this  case  the  sum  of  each  vertical  couple  is  16,  and  there  are 
8  couples ;  hence  the  sum  of  these  8  couples  is  8  x  16  =  128. 
And  in  general  the  sum  will  be  the  product  of  one  of  the 
couples  into  the  number  of  couples.  It  will  be  observed 
that  the  first  couple  contains  the  first  and  last  terms,  I  and 
15  ;  therefore  the  sum  of  the  double  series  is  equal  to  the 
product  of  the  sum  of  the  first  and  last  terms  into  the 
number  of  terms.  Or  if  5  be  put  to  represent  the  sum  of 
the  series,  we  shall  have— 

2  S  =  (a  +  I)  n, 
and,  dividing  both  sides  by  2— 

-S  =  („  +  /)£;  (I23.) 


GEOMETRICAL  PROGRESSION   EXPLAINED.  435 

Or,  in  words :  The  sum  of  an  arithmetical  series  equals  the  prod- 
uct of  the  sum  of  the  first  and  last  terms,  into  half  the  number 
of  terms. 

430. — Geometrical  Progression. — A  series  of  numbers, 
such  as  i,  2,  4,  8,  16,  32,  64,  128,  256,  etc.,  in  which  any  one 
of  the  terms  is  obtained  by  multiplying  the  preceding  one 
by  a  constant  quantity,  is  termed  a  Geometrical  Progression. 
The  constant  quantity  is  termed  the  common  Ratio,  and  is 
equal  to  any  term  divided  by  the  preceding  one.  Thus  in 

the  above  example  --  or  —  Or  —  —  2,  equals  the  common  ra- 

o  42 

tio  of  the  above  series.  In  the  series,  i,  3,  9,  27,  etc.,  we 
have  for  the  ratio— 

27  _  9  _  3 
~9   "3^1  : 

which  is  the  common  ratio  of  this  series. 
A  geometrical  series  may  be  put  thus : 

Terms:  I,     2,         3,  4; 

Progress. :  i,     i  x  3,   i  x  3  x  3,   i  x  3  x  3  x  3  ; 

or  thus — 

Terms:  I,    2,          3,          4: 

Progress. :  i,    i  x  3,   i  x  3 2,  i  x  3 8 ; 

in  which  the  common  ratio,  in  this  case  3,  appears  in  each 
term  and  with  an  exponent  which  is  equal  to  the  number  of 
terms  preceding  that  in  which  it  occupies  a  place. 

If  the  first  term  be  represented  by  a  and  the  common  ra- 
tio by  r,  then  the  following  will  represent  any  geometrical 
progression — 

a,  ar,  arz,  ar3,  #r4,  etc.  (124.) 

For  example,  let  a  =  2  and  r  =  4 ;  then  the  progression 
will  be — 

2,  8,  32,   128,  512,  etc. 


436  ALGEBRA. 

If  r  =  unity,  then  when  a  =  2  the  progression  becomes  — 
2,  2,  2,  2,  2,  etc. 

If  r  be  less  than  unity,  then  the  progression  will  be  a  de- 
creasing one. 

For  example,  let  a  •=.  2  and   r  =  -J.     Then  we  have  for 
the  progression  — 

1  1   L  JL 

2j  Iy  2'  4'  8'  16'  e 

If  the  number  of  terms  be  represented  by  n,  and  the  last  by 
/,  then  the  last  term  will  be  — 


For  example,  let  n  equal  6,  then  the  progression  will  be  — 

Terms:  i,         2,         3,         4,         5,         6; 

Progress.:        a,       ar,       ar*y     ar3,    ar\     ar*\ 

in    which   the  .exponent   of   the   last  term  equals  n  —  i  = 
6  _  i  =  5. 

If  5  be  put  for  the  sum  of  a  geometrical  progression,  we 
will  have  — 

S  =  a  +  ar  +  arz  +  .....  +  a  rn~*  +  a  rll~l. 
Multiply  each  member  by  ry  then— 

Sr  =  ar  +  ar2  +  .....  +  a  rn~2  +  a  r*~l  +  arn. 
Subtract  the  upper  line  from  the  lower;  then  — 

S  r  =  ar  +  ar2  +  .....  +  arn~2  +  a  r11'1  +  a  r*, 

S     =g  +  ar+ar2+  .....  +  arn~2  +  ar*~l  _ 
Sr  -  s  =  -  a     *"*  *          ~*         +  ar", 

Sr  —  s  =  —  a  +  arn, 

S  (r  —  i)  =  —  a  +  a  rn  =  arn  —  a, 

ar"  ~~  a 


GEOMETRICAL  PROGRESSION   CONTINUED.  437 

The  last  term  (equation  (125.))  equals  I—  arn~l,  and   since 
arn  =  r  x  arn~  *  =  r  /,  therefore— 


Thus,  to  find  the  sum  of  a  geometrical  progression  :  Multi- 
ply the  last  term  by  the  ratio  ;  from  tJie  product  deduct  the  first 
term,  and  divide  the  remainder  by  the  ratio  less  unity. 

For  example,  the  sum  of  the  geometrical  progression  — 

•S  =  i  +  3  +  9  +  27  +  81  +  243  +729  =  1093 

by  actual  addition. 

To  obtain  it  by  the  above  rule  — 


rl  —  a       3  x  729  —  I 

s==—  — :    TfST    II093' 

the  correct  result. 

If  there  be  a  decreasing  geometrical  progression,  as  i,  -J, 
•J,  ^T,  etc.,  in  which  the  ratio  equals  ^,  the  sum  will  be — 

S=i    »  T  +  "  +  ^  +  -g7>  etc->  to  infinity. 
Multiply  this  by  3,  and  subtract  the  first  from  the  last — 


35=3+1+-  +  -  +  —  +  —  +  to  infinity. 

S=  i  +i  +  i  +  ^  +  gIf  + to  infinity. 

2  5  =  3»  or  5  =  if 

In  a  decreasing  progression  let  r,  the  common  ratio,  be 
represented  by  —  (b  less  than  c),  and  the  first  term  by  a,  then 
ihe  sum  will  be — 

b          b2         b3 
S=a  +  a-  +  a-  +  a-3+,  etc.,  to  infinity. 


438  ALGEBRA. 

Multiply   this   by  -,  and  subtract  the  product   from    the 


above  — 


^b          b          d*         b* 
S-  =  a-  +  a-  +  a-^  +  etc.,  to  infinity. 
c  c          c          c 

b          b*          l>3 
=z  a  +  a  —  +  a—2  +  a-j  +  to  infinity. 

£  b          b*         b3 

S  —  —  a  —  \-  a  —  +  <2  -j  +  to  infinity. 
cere  * 


Or—  s(i  --)«*, 


For  example,  let  the  first  term  of  a  geometrical  progression 
equal  2,  and  the  ratio  equal  £,  then  the  sum  will  be— 


From  this,  therefore,  we  have  this  rule  for  the  sum  of  an  in- 
finite geometrical  progression,  namely  :  Divide  the  first  term 
by  unity  less  the  ratio. 


SECTION  X.— POLYGONS. 

431. — Relation  of  Sum  and  Difference  of  Two  Lines. — 

Let  AB  and  CD  (Fig.  285)  be  two  given  lines;  make  EH 


-B 
D 


E  -  ]  —  |- 
H     J 

FIG.  285. 

equal  to  A  B,  and  HG  equal  to  CD-,  then  E  G  equals  the 
sum  of  the  two  lines. 

Make  FG  equal  to  A  B,  which  is  equal  to  EH. 

Bisect  E  G  in  J  ;  then,  also,  J  bisects  HF\  for— 


and  — 

EH=FG. 

Subtract  the  latter  from  the  former  ;  then  — 

EJ- 
but— 

E 

and  — 
therefore  — 


Now,  E  J  is  half  the  sum  of  the  two  lines,  and  HJ  is  half 
the  difference  ;  and— 

Ey-Hy=EH=AB. 

Or  :  Half  the  sum  of  two  quantities,  minus  half  their  dif- 
ference, equals  the  smaller  of  the  two  quantities. 


440 


POLYGONS. 


Let  the  shorter  line  be  designated  by  a,  and  the  longer 
by  b ;   then  the  proposition  is  expressed  by — 


_a+b      b  —  a 

2  2 


(128.) 


We  also  have  EJ+JF—EF—  CD\  or,  half  the  mm 
of  two  quantities,  plus  half  their  difference,  equals  the  larger 
quantity. 

432. — Perpendicular,  in  Triangle  of  Known  Side§. — 

Let  ABC  (Fig.  286)  be  the  given  triangle,  and  CE  a  perpen- 
dicular let  fall  upon  A  B,  the  base.     Let  the  several  lines  of 


the  figure  be  represented  by  the  symbols  a,  b,  c,  d,  g,  and  f, 
as  shown.  Then,  since  A  EC  and  BEC  are  right-angled 
triangles,  we  have  (Art.  416)  the  following  two  equations, 
and,  by  subtracting  one  fr.om  the  other,  the  third— 


Then  (Art.  414),  by  substitution,  we  have  — 

(f  +  f)(f-e)  =  («  +  *)(«-*)• 

By  division  we  obtain  — 


_ 


a-  6) 


f+g 


RULE   FOR  TRIGONS.  441 

According  to  Art.  431,  equation  (128.),  we  have — 


In  this  expression  let  the  value  of  /  —  g,  as  above,  be 
substituted,  then  we  will  have— 


~=f+*       (<*  +  *)  (a  -  b) 


Multiply  the  first  fraction  by  (f  +  g),  then  join  the  two 
fractions,  when  we  will  have  — 


The  lines  f  and  g,  in  the  figure,  together  equal  the  line 
c  ;  therefore,  by  substitution  — 

f  -  (a  +  b)  (a  -  b] 
g  =  -^-        -i.  (129.) 

This  is  the  value  of  the  line  g. 

It  may  be  expressed  in  words,  thus:  The  shorter  of  the 
two  parts  into  which  the  base  of  a  triangle  is  divided  by  a 
perpendicular  let  fall  from  the  apex  upon  the  base,  equals  the 
quotient  arising  from  a  division  by  twice  the  base,  of  the  differ- 
ence between  the  square  of  the  base  and  the  product  of  the  sum 
and  difference  of  the  two  inclined  lines. 

As  an  example  to  show  the  application  of  this  rule,  let 
a  —  9,  b  =  6,  and  c  =  12  ;  then  equation  (129.)  becomes  — 

12'  -  (9  +  6)  (9  -  6) 

2  X  12 

.  .144  -  iT><"3 

- 


99- 


442  POLYGONS. 

Now,  to  obtain  the  length  of  d,  the  perpendicular,  by  the 
figure,  we  have  — 


-and,  extracting  the  square  root  — 


or,  in  words :  The  altitude  of  a  triangle  equals  the  square  root 
of  the  difference  of  the  squares  of  one  of  the  inclined  sides  and 
its  base. 

As  an  example,  take  the  same  dimensions  as  before,  then 
equation  (130.)  becomes — 


The  square  of  6    =  36- 

"     4i  =  17-015625 

62-4i2=  18^984375, 
the  square  root  of  which  is  4-44234;  therefore 


d=  t--  4^  =  4. 44234. 

This  may  be  tested  by  applying  the  rule  to  the  other  in 
clined  side  and  its  base — 

c  =  12 
*  =    4* 
/=    71- 


Then,  ^- 

9'  =  81- 

?%*=  62-015625 
9'  -  7F  =  ^8-984375. 


TRIGON — RADIUS   OF   CIRCLES. 


443 


The  same  result  as  before,  producing  for  its  square  root 
the  same,  4-44234,  the  value  of  d\  therefore— 


433.  —  Trigon  :  Radiu§  of  €ircum§cribed  and  In§cribed 
Circles:  Area.—  Let  A  B  C  (Fig.  287)  be  a  given  trigon  or 
triangle  with  its  circumscribed  and  inscribed  circles.  Draw 
the  lines  A  D  F,  DB  and  D  C. 

The  three  triangles,  A  B  D,  A  CD,  and  B  D  C,  have  their 
apexes  converging  at  D,  and  form  there  the  three  angles, 
A  DB,  ADC,  and  B  D  C.  These  three  angles  together  form 


four  right  angles  (Art.  335),  and   each  of  them,  therefore, 
equals  f  of  a  right  angle. 

The  angles  of  the  triangle  BDC  together  equal  two 
right  angles  (Art.  345).  As  above,  the  angle  BDC  equals  | 
of  a  right  angle,  hence  2  —  -J  =  ^^  —  |  of  a  right  angle, 
equals  the  sum  of  the  two  remaining  angles  at  B  and  C. 
The  triangle  BDC  is  isoceles  (Art.  338);  for  the  two  sides 
B  D  and  D  C,  being  radii,  are  equal ;  therefore  the  two  angles 
at  the  base  B  and  C  are  equal,  and  as  their  sum,  as  above, 
equals  f  of  a  right  angle,  therefore  each  angle  equals  -J  of  a 
right  angle.  Draw  the  two  lines  FC  and  F B.  Now,  be- 


444  POLYGONS. 

cause  Z>  C  and  DF  are  radii,  they  are  equal,  hence  DFC  is 
an  isoceles  triangle. 

It  was  before  shown  that  the  angle  B  D  C  equals  |  of  a 
right  angle;  now,  since  the  diameter  A  F  bisects  the  chord 
B  C,  the  angles  B  D  E  ^nd  E  D  C  are  equal,  and  each  equals 
the  half  of  the  angle  B  D  C\  or,  \  of  f  of  a  right  angle  equals 
£  of  a  right  angle.  Deducting  this  from  two  right  angles 
(the  sum  of  the  three  angles  of  the  triangle),  or  2  —  f  = 
\\  —  |  of  a  right  angle  equals  the  sum  of  the  angles  at  F  and 
C',  hence  each  equals  the  half  of  f,  or  f  of  a  right  angle; 
therefore  the  triangle  DFC  is  equilateral.  The  triangles 
DBF  and  DFC  are  equal.  The  angles  B  D  C  and  B  F  C  are 
equal;  the  line  BC  is  perpendicular  to  D  F  and  bisects  it, 
making  DE  and  EF  equal;  hence  DE  equals  half  D  F,  or 
DB,  radii  of  the  circumscribing  circle.  Therefore,  putting 
R  to  represent  B  D,  the  radius  of  the  circumscribing  circle, 
and  b  =  B  C,  a  side  of  the  triangle  A  B  C,  by  Art.  416,  we 
have — 

+  DE 


Transferring  and  reducing— 


4  "    4' 


4/       4 
Ijpt^i^ 

4         ~4 

4  x  I^»==i3»==^ 
34        ~3        ~3  ' 


Or,  The  Radius  of  the  circumscribing  circle  of  a  regular  trigon 
or  equilateral  triangle,  equals  a  side  of  the  triangle  divided  by 
the  square  root  of  3. 


AREA   OF   EQUILATERAL  TRIANGLE. 


445 


By   reference  to  Fig.  287  it   will  be  observed,   as   was 
above  shown,  that  D  E  =  E  F=  -  =  -  ;  or,  D  E,  the  ra- 

dius of  the  inscribed  circle,  equals  half  the  radius  of  the 
circumscribed  circle;  or,  again,  dividing  equation  (131.)  by 
2,  we  have  — 

R  _        b 

2  "  2  t/y- 

and,  putting  r  for  the  radius  of  the  inscribed  circle,  we 
have  — 


Or:  The  radius  of  the  inscribed  circle  of  a  regular  trigon  equals 
the  half  of  a  side  of  the  trigon  divided  by  the  square  root  of  3. 
To  obtain  the  area  of  a  trigon  or  equilateral  triangle  ;  we 
have  (Art.  408)  the  area  of  a  parallelogram  by  multiplying 
its  base  into  its  height  ;  and  (Arts.  341  and  342)  the  area  of  a 
triangle  is  equal  to  half  that  of  a  parallelogram  of  equal  base 
and  height,  therefore,  the  area  of  the  triangle  BD  &(Fig.  287) 
is  obtained  by  multiplying  B  C,  the  base,  into  the  half  of 
ED,  its  height.  Or,  when  A^  is  put  for  the  area  — 


or_  ,:;/;'•  *=>*£., 

substituting  for  R  its  value  (131.)—" 

jr=*x-4= 

4  1/3 


4^3 

This  is  the  area  of  the  triangle  BD  C. 

The  triangle  A  B  C  is  compounded  of  three  equal  tri- 
angles, one  of  which  is  the  triangle  B  D  C ;  therefore  the 
area  of  the  triangle  ABC  equals  three  times  the  area  of  the 
triangle  B DC\  or,  when  A  represents  the  area — 


446 


POLYGONS. 


4  1/3 


(1330 


Or:  The  area  of  a  regular  /rz^wz  or  equilateral  triangle 
equals  three  fourths  of  the  square  of  a  side  of  the  triangle  di- 
vided by  the  square  root  of  3. 


—  Tetragon  ;  Radius  of  Circumscribed  and  In- 
scribed Circles:  Area.  —  Let  A  B  CD  (Fig.  288)  be  a  given 
tetragon  or  square,  with  its  circumscribed  and  inscribed 


FIG.  288. 

circles,  of  which  A  E  is  the  radius  of  the  former  and  EF  that 
of  the  latter.  The  point  F  bisects  A  B,  the  side  of  the 
square.  A  F  equals  EF  and  equals  half  A  B,  a  side  of  the 
square.  Putting  R  for  the  radius  of  the  circumscribed 
circle  and  b  for  A  B,  we  have  (Art.  416)— 


:7T:  <'34-) 

Or:  The  radius  of  the  circumscribed  circle  of  a  regular  tetra- 
gon equals  a  side  of  the  square  divided  by  the  square  root  of  2. 


SIDE   AND   AREA   OF   HEXAGON. 


447 


By  referring  to  the  figure  it  will  be  seen  that  the  radius 
of  the  inscribed  circle  equals  half  a  side  of  the  square — 


b 


r  — 


(135.) 


The.  area  of  the  square  equals  the  square  of  a  side — 

A  =  b*.  (136.) 

435.  —  Hexagon:  Radius  of  Circumscribed  and  In- 
scribed Circles:  Area.— Let  A  B  C  D  EF(Fig.  289)  be  an  equi- 
lateral hexagon  with  its  circumscribed  and  inscribed  circles, 
of  which  EG  is  the  radius  of  the  former,  and  G H that  of 
the  latter.  The  three  lines,  A  D,  BE,  and  C  F,  divide  the 


FIG.  289. 

hexagon  into  six  equal  triangles  with  their  apexes  converg- 
ing at  G.  The  six  angles  thus  formed  at  G  are  equal,  and 
since  their  sum  about  the  point  G  amounts  to  four  right 
angles  (Art.  335),  therefore  each  angle  equals  £  or  f  of  a 
right  angle.  The  sides  of  the  six  triangles  radiating  from  G 
are  the  radii  of  the  circle,  hence  they  are  equal ;  therefore, 
each  of  the  triangles  is  isosceles  (Art.  338),  having  equal  angles 
at  the  base.  In  the  triangle  EGD,  the  sum  of  the  three 
angles  being  equal  to  two  right  angles  (Art.  345),  and  the 
angle  at  G  being,  as  above  shown,  equal  to  f  of  a  right  angle, 
therefore  the  sum  of  the  two  angles  at  E  and  D  equals 
2  —  £  =  J  of  a  right  angle  ;  and,  since  they  equal  each  other, 


44^  POLYGONS. 

therefore  each  equals  f  of  a  right  angle  and  equals  the  angle 
at  G ;  therefore  E  G  D  is  an  equilateral  triangle.  Hence 
ED,  a  side  of  a  hexagon,  equals  E  G,  the  radius  of  the  circum- 
scribing circle — 

R=b.  (137.) 

As  to  the  radius  of  the  inscribed  circle,  represented  by  G  H, 
a  perpendicular  from  the  centre  upon  ED,  the  base;  the 
point  H  bisects  E  D.  Therefore,  E  H  equals  half  of  a  side 
of  the  hexagon,  equals  half  the  radius  of  the  circumscribing 
circle.  Let  R  =  this  radius,  and  r  the  radius  of  the  inscribed 
circle,  while  b  =  a  side  of  the  hexagon  ;  then  we  have  (Arts. 
353  and  416)— 


i 

r  — 


Now,  R  =1  b,  therefore — 


r=^--      '  (138.) 

Or :  The  radius  of  the  inscribed  circle  of  a  regular  hexagon 
equals  the  half  of  a  side  of  the  hexagon,  multiplied  by  the 
square  root  of  3. 

As  to  the  area  of  the  hexagon,  it  will  be  observed  that  the 
six  triangles,  A  B  G,  B  G  C,  etc.,  converging  at  G,  the  centre, 
are  together  equal  to  the  area  of  the  hexagon.  The  area  of 
E  G  D,  one  of  these  triangles,  is  equal  to  the  product  of  £  D, 
the  base,  into  the  half  of  G  H,  the  perpendicular ;  or,  when 
N  is  put  to  equal  the  area— 

G  H 


SIDE  AND  AREA   OF  OCTAGON. 


449 


and,  since  rt  as  above,  equals   *  3  —  , 

2 


This  is  the  area  of  one  of  the  six  equal  triangles  ;  therefore, 
when  A  is  put  to  represent  the  area  of  the  hexagon,  we  have— 


A  = 


(139.) 


Or :  The  area  of  a  regular  hexagon  equals  three  lialves  of  the 
square  of  a  side  multiplied  by  the  square  root  of  $. 


FIG.  290. 

4-36- — Octagon:  Radius  of  Circumscribed  and  In- 
icribed  Circles:  Area. — Let  C E D B F (Fig.  290)  represent  a 
quarter  of  a  regular  octagon,  in  which  ^is  the  centre,  ED 
a  side,  and  CE  and  DB  each  half  a  side,  while  CF  and 
£Fare  radii  of  the  inscribed  circle,  and  BF  and  DF  are 
radii  of  the  circumscribed  circle. 


450  POLYGONS. 

Let  R  represent  the  latter,  and  r  the  former ;  also  let  b 
represent  ED,  one  of  the  sides,  and  n  be  put  for  A  D,  and 
for  A  E.  Then  we  have — 


b_ 

2~    T* 
b 

or-  n  =  r--> 

Since  A  D  Eis  a  right-angled  triangle  (Art.  416),  we  have — 

T  =  ED\ 

n*  =  b\ 
b* 


Placing  the  value  of  n,  equal  to  the  value  before  found, 
we  have — 


b       b 


f  i         i\ 
-=f  —  -  +  -  ) 

2        V  1/2         2/ 


This  coefficient  may  be  reduced  by  multiplying  the  first 
fraction  by   ^2,  thus— 

JL  x  t5  =  .^i 

V2   X  2     X> 


RULES  FOR  OCTAGON.  451 

therefore — 

r  = 


Or  :  The  radius  of  the  inscribed  circle  of  a  regular  octagon 
equals  half  a  side  of  the  octagon  multiplied  by  the  sum  of  unity 
plus  the  square  root  of  2.  In  regard  to  the  radius  of  the  cir- 
cumscribed circle,  by  Art.  416  we  have  — 


In  this  expression  substituting  for  ra,  its  value  as  above,  we 
have  — 


The  square  of  the  coefficient  (  t/2  +  i  )  by  Art.  412  equals 
21/2+1  =21/2  +  3,  then  — 


Or :  The  radius  of  the  circumscribed  circle  of  a  regular  octagon 
equals  half  a  side  of  the  octagon  multiplied  by  the  square  root  of 
the  sum  of  twice  the  square  root  of  2  plus  4. 

In  regard  to  the  area  of  the  octagon,  the  figure  shows 
that  one  eighth  of  it  is  contained  in  the  triangle  D  E  F. 


452  POLYGONS. 

The  area  of  D  E  F,  putting  it  equal  to  N,  is  — 

B  F 
N  =  EDx  —  -, 


N  =  b  x  —  , 


AT  =  (  1/2"  +  i)  —  . 
4 

This  is  the  area  of  one  eighth  of  the  octagon  ;  the  whole 
area,  therefore,  is  — 


. 

4 
A  =  (V~2+i)2b\  (142.) 

Or  :  The  area  of  a  regular  octagon  equals  twice  the  square  of  a 
side,  multiplied  by  the  sum  of  the  square  root  of  2  added  to  unity. 
When  a  side  of  the  enclosing  square,  or  diameter  of  the 
inscribed  circle,  is  given,  a  side  of  the  octagon  may  be  found  ; 
for  from  equation  (140.),  multiplying  by  two,  we  have  — 

2  r  -  (  V~2  +  i)  b. 
Dividing  by  V.2  +  i,  gives  — 


The  numerator,  2  r,  equals  the  diameter  of  the  inscribed 
circle,  or  a  side  of  the  enclosing  square  ;  therefore  : 

The  side  of  a  regular  octagon,  equals  a  side  of  the  enclosing 
square  divided  by  the  sum  of  the  square  root  of  2  added  to  unity. 

437.—  Dodecagon  :  Radius  of  Circumscribed  and  In- 
scribed Circles:  Area.  —  Let  A  B  C  (Fig.  291)  be  an  equilat- 


SIDE  AND  AREA  OF  DODECAGON.  453. 

era!  triangle.  Bisect  A  B  in  F\  draw  C FD  ;  with  radius  A  C 
describe  the  arc  A  D  B.  Join  A  and  D,  also  D  and  B  ;  bisect 
A  D  in  E  ;  with  the  radius  E  C  describe  the  arc  E  G.  Then 
A  D  and  D  B  are  sides  of  a  regular  dodecagon,  or  twelve- 
sided  polygon  ;  of  which  A  C,  D  C,  and  B  C  are  radii  of  the 
circumscribing  circle,  while  E  C  is  a  radius  of  the  inscribed 
circle. 

The  line  A  B  is  the  side  of  a  regular  hexagon  (Art.  435). 
Putting  R  equal  to  A  C  the  radius  of  the  circumscribing  cir- 
cle ;  r,  =  E  C,  the  radius  of  the  inscribed  circle  ;  £,  =  A  D,  a 
side  of  the  dodecagon,  and  n  —  D  F.  Then  comparing  the 


FIG.  291. 

homologous  triangles,  ADF  and  A  EC  (the  angle  ADF 
equals  the  angle  EA  C,  and  the  angles  DFA  and  A  E  C  are 
right  angles);  therefore,  the  two  remaining  angles  DAF 
and  A  CE  must  be  equal,  and  the  two  triangles  homologous 
(Art.  345).  Thus  we  have— 

DF  :  DA   :  :  AE  :  A  C, 
n  :  b  :  :      :  R, 


**& 


454  POLYGONS. 

In  Art.  435  it  was  shown  that  FC  (Fig.  291),  or  G  H  oi 
Fig.  289,  the  radius  of  the  inscribed  hexagon,  equals  V~l  ~, 

n 

and  in  which  its  b  =  R  ;  Fc  —  VJ  —  . 

Now  (  Fig.  291)  — 

=  DC  -  FC, 


or  — 

n  =  R-^~ 

Substituting  this  value  of  n,  in  the  above  expression,  we 
have  — 

R- 
~ 

Multiplying  by  R  and  reducing,  we  have 


R  =  |_  ^_  b.  (144.) 


Or  :  The  radius  of  the  circumscribed  circle  of  a  regular  dodec- 
agon, equals  .  a  side  of  the  dodecagon  multiplied  by  the  square 
root  of  a  fraction,  having  unity  for  its  numerator  and  for  its 
denominator  2  minus  the  square  root  of  3. 

Comparing  the  same  triangles,  as  above,  we  have  — 

FD  \  FA  :  :  EA  ;  EC, 
or— 


R 

.  b 

n  :  — 

2 

:  :     -  :  r, 

Rb 

Rb 

~  4  n  ~  * 

\R(\  -i|/^ 

b 

('450 


RULE   FOR  DODECAGON.  455 

Or  :  The  radius  of  the  inscribed  circle  of  a  regular  dodecagon 
equals  a  side  of  the  dodecagon  divided  by  the  difference  between 
4  and  the  square  root  of  3. 

The  area  of  a  dodecagon  is  equal  to  twelve  times  the  area 
of  the  triangle  ADC  (Fig.  291).  The  area  of  this  triangle  is 
equal  to  half  the  base  by  its  perpendicular  ;  or,  A  E  x  E  C  ; 
or  — 

b 


or,  where  N  equals  the  area  — 


Or,  for  the  area  of  the  whole  dodecagon  — 

12  N  —  6  br, 
A  =6br. 

Substituting  for  r  its  value  as  above,  we  have  — 


Or  :  The  area  of  a  regular  dodecagon  equals  the  square  of  a 
side  of  the  dodecagon,  multiplied  by  a  fraction  having  6  for  its 
numerator,  and  for  its  denominator,  4  minus  twice  the  square 
root  of  3. 

438.  —  Hecadecagon  :  Radius  of  Circumscribed  and  Iii- 
§cribed  Circle§  :  Area.  —  Let  A  B  CD  (Fig.  292)  be  a  square 
enclosing  a  quarter  of  a  regular  octagon  C  EFB,  E  F  being 
one  of  its  sides,  and  C  E  and  FB  each  half  a  side,  while  F  D 
is  the  radius  of  the  circumscribed  circle,  and  J  D  the  radius 
of  the  inscribed  circle  of  the  octagon.  Draw  the  diagonal 
A  D  ;  with  DFior  radius,  describe  the  circumscribed  circle 
EGF\  join  G  with  F  and  with  E  ;  then  EG  and  GFvfill 
each  be  a  side  of  a  regular  hecadecagon,  or  polygon  of  six- 
teen sides. 

An  expression  for  F  D,  the  radius  of  the  circumscribed 


456 


POLYGONS. 


circle,  may  be  obtained  thus:  Putting  FD  =  R]  H D  =  r; 

G F  =  b\  GJ  —  n\  and  J F  —  -  (Art.  416),  we  have — 

GT  =  GF*  —  JF\ 

••  =  »•-  (9- 


C  D 

FIG.  292. 

Comparing  the  two  homologous  (Art.  361)  triangles,  GJF 
and  F  H  D  (Art.  374),  we  have — 

Gy  :  GF  :  :  HF  :  FD, 
n,     b     ::      I    :  *, 


Putting  this  value  of  n'  in  an  equation  against  the  former 
value,  we  have— 


In  Art.  436,  the  value  of  F  D,  as  the  radius  of  the  cir- 
cumscribed circle  of  a  regular  octagon,  is  given  in  equation 
(141.)  as— 

b 


R 


V2 


SIDE  AND  AREA   OF  HECADECAGON.  457 

in  which  b  represents  a  side  of  the  octagon,  or  E  Ft  for 
which  we  have  put  s.  Substituting  s  for  b  and  putting  the 
numerical  coefficient  under  the  radical,  equal  to  B,  we 
have  — 


Squaring  each  member  gives— 


From  which,  by  transposition,  we  have  — 


£2 

Substituting  in  the  above  expression  for  (—  )  ,  this  value 

•  \2  / 

of  it,  gives  — 

^-  =  t>>-*\ 
4^2~  B     . 

Transposing,  we  have— 

-*1  +  *'=*-. 

4R*      B 

Multiplying  the  first  term  by  B,  and  the  second  by 
we  have  — 

* 


Bb*  +  4^4  _  ^ 


Transposing,  we  have  — 

2=  -Bb\ 


458  POLYGONS. 

To  complete  the  square  (Art.  428)  we  proceed  thus  — 


Taking  the  square  root,  we  have 


Restoring  B  to  its  value,  2  I/I  +  4  as  above,  we  have— 


B  —  I  =  2^2 

multiply  these  — 


2  +  2  i/J, 

3  + 


=  4/2"+  2. 
Therefore— 


2.  (147.) 


RULES   FOR   HECADECAGON.  459 

Or:  The  radius  of  the  circumscribed  circle  of  a  regular 
hecadecagon  equals  a  side  of  the  hecadecagon  multiplied  by  the 
square  root  of  the  sum  of  two  quantities,  one  of  which  is  the 
square  root  of  2  added  to  2,  and  the  other  is  the  square  root  of 
the  sum  of  seven  halves  of  the  square  root  of  2  added  to  5. 

To  obtain  the  radius  of  the  inscribed  circle  we  have  (Fig. 
292)— 

H  D*  =  FD*  —  H  F\ 


Substituting  for  R  a  its  value  as  above,  we  have  — 


r  a  =  &*  (       B(B}  +  *  B)  -          , 


The  coefficient  of  b  is  the  same  as  in  the  case  above,  ex- 
cept the  —  i;  therefore  its  numericaK  value  will  be  i  less, 
or— 


r  =  b  \/  ^S  +  I  V2  +  V2  +  if.  (148). 

Or:  The  radius  of  the  inscribed  circle  of  a  regular  hecadeca- 
gon equals  a  side  of  the  hecadecagon  multiplied  by  the  square  root 
of  two  quantities,  one  of  which  is  the  square  root  of  2  added  to 
if,  and  the  other  is  the  square  root  of  the  sum  of  seven  halves  of 
the  square  root  of  2  added  to  5. 

To  obtain  the  area  of  the  hecadecagon  it  will  be  observed 
that  the  area  of  the  triangle  G  FD  (Fig.  292)  equals  HD  x 
H F,  and  that  this  is  the  TV  part  of  the  polygon ;  we  there- 
fore have — 

A  =  \6HDxHF, 
A  =  i6r-  =  Sr&. 

2 


460  POLYGONS. 

The  value  of  r  is  shown  in  (148.);  therefore  we  have — 


A  =  8  b 


+  1  . 


+  if. 


(H9-) 


Or :  The  area  of  a  regular  hecadecagon  equals  eight  times 
the  square  of  its  side,  multiplied  by  the  square  root  of  two  qtian- 
tities,  one  of  which  is  the  square  root  of  2  added  to  if,  and  the 
other  is  the  square  root  of  the  sum  of  seven  halves  of  the  square 
root  of  2  added  to  5. 

439.— Polygon*  :  Radius  of  Circumscribed  and  I  u  scribed 
Circles :  Area. — In  Arts.  433  to  438  the  relation  of  the  radii 
to  a  side  in  a  trigon,  tetragon,  hexagon,  octagon,  dodeca- 
gon and  hecadecagon  have  been  shown  by  methods  based 


upon  geometrical  proportions.  This  relation  in  polygons 
of  seven,  nine,  ten,  eleven,  thirteen,  fourteen  and  fifteen 
sides,  cannot  be  so  readily  shown  by  geometry,  but  can  be 
easily  obtained  by  trigonometry — as  also  said  relation  of  the 
parts  in  a  regular  polygon  of  any  number  of  sides.  The  na- 
ture of  trigonometrical  tables  is  discussed  in  Arts.  473  and 
474.  So  much  as  is  required  for  the  present  purpose  will 
here  be  stated. 

Let  ABC  (Fig.  293)  represent  one  of  the  triangles  into 
which  any  polygon  may  be  divided,  in  which  B  C  =  b  =  a 
side  of  the  polygon ;  A  C  —  R  =  the  radius  of  the  circum- 
scribed circle  ;  and  A  D  =  r  =  the  radius  of  the  inscribed 
circle. 


GENERAL  RULES  FOR  POLYGONS.          461 

Make  E  C  equal  unity  ;  on  C  as  a  centre  describe  the  arc 
E  F\  draw  FH  and  E  G  perpendicular  to  B  C,  or  parallel  to 
A  D ;  then  for  the  uses  of  trigonometry  E  G  is  called  the 
tangent  of  c,  or  of  the  angle  A  C  B,  and  FHis  the  sine,  and 
H  C  the  cosine  of  the  same  angle. 

These  trigonometrical  quantities  for  angles  varying  from 
zero  up  to  ninety  degrees  have  been  computed  and  are  to  be 
found  in  trigonometrical  tables. 

Referring  now  to  Fig.  293  we  have — 

HC  :  FC  :  :  DC  :  AC, 

b 
cos.  c  :    I   :  :   —  :  R, 

(150.) 

Again — 

E  C  :  E  G  :  :  D  C  :  A  D, 

b 
I  :  tan.  c  :  :  -  :  r, 

r  =  -•  tan.  c.  (151.) 

These  two  equations  give  the  required  radii  of  the  cir- 
cumscribed and  inscribed  circles.  They  may  be  stated  thus  : 

The  radius  of  the  circumscribed  circle  of  any  regular  poly- 
gon equals  a  side  of  the  polygon  divided  by  twice  the  cosine  of 
the  angle  formed  by  a  side  of  the  polygon  and  a  radius  from  one 
end  of  the  side. 

The  radius  of  the  inscribed  circle  of  any  regular  polygon 
equals  half  of  a  side  of  the  polygon  imiltiplied  by  the  tangent  of 
the  angle  formed  by  a  side  of  the  polygon  and  a  radius  from  one 
end  of  the  side. 

The  area  of  a  polygon  equals  the  area  of  the  triangle 
ABC  (Fig.  293),  (of  which  B  C  is  one  side  of  the  polygon 
and  A  is  the  centre),  multiplied  by  the  number  of  sides  in 
the  polygon ;  or,  if  n  be  put  to  represent  the  number  of  the 
sides  and  A  the  area,  then  we  have— 


POLYGONS. 

A  =  Bn, 


in  which  B  equals  the  area  of  the  triangle.     The  area  of  A 
B  C  (Fig.  293)  is  equal  to  AD  x  B  D,  or  — 


For  r  substituting  its  value,  as  in  equation  (151.),  we  have  — 

b  b        i  7 

B  =  -  tan.  c-  =  —  b"  tan.  c. 
2  24 

Therefore,  by  substitution  — 

A  =-b*n  tan.*.  (152.) 

Or  :  The  area  of  a  regular  polygon  equals  the  square  of  a 
side  of  the  polygon,  multiplied  by  one  fourth  of  the  number  of 
its  sides,  and  by  the  tangent  of  the  angle  formed  by  a  side  of  the 
polygon,  and  a  radius  from  one  end  of  the  sides. 

440.  —  Polygons  :  Their  Angles.  —  Let  a  line  be  drawn 
from  each  angle  of  a  regular  polygon  to  its  centre,  then 
these  lines  form  with  each  other  angles  at  the  centre,  which 
taken  together  amount  to  four  right  angles,  or  to  360  de- 
grees (Arts.  327,  335). 

If  this  360  degrees  be  divided  by  the  number  of  the  sides 
of  the  polygon,  the  quotient  will  equal  the  angle  at  the  cen- 
tre of  the  polygon,  of  each  triangle  formed  by  a  side  and  two 
radii  drawn  from  the  ends  of  the  side.  For  example:  if 
ABC  (Fig.  293)  be  one  of  the  triangles  referred  to,  having 
B  C  one  of  the  sides  of  the  polygon  and  the  point  A  the  cen- 
tre of  the  polygon,  then  the  angle  B  A  (Twill  be  equal  to  360 
degrees  divided  by  the  number  of  the  sides  of  the  polygon. 
If  the  polygon  has  six  sides,  then  the  angle  B  A  C  will  contain 

=  60  degrees  ;  or  if  there  be  10  sides,  then  the  angle  at 
A,  the  centre,  will  contain  -  ----  =   36   degrees.      The  angle 


SIDE  AND   AREA   OF   PENTAGON.  463 

BAD  equals  half  the  angle  B  A  C,  or,  when  n  equals  the 
number  of  sides,  the  angle  BAG  equals — 

360 

» 

n 

B  A  C 
and  the  triangle  B  A  D  = ,  equals — 

360 

2  n 

Now  the  angles  B  A  D  +  D  B  A   equal  one  right   angle 
(Art.  346),  or  90  degrees.     Hence  the  angle  DBA  =90°  - 
BAD,or  the  angle  c  equals — 

(1530 


2  n 


For  example,  if  n  equal  6,  or  the  polygon  have  six  sides, 
then— 


Therefore,  the  angle  c,  contained  in  equations  (150.),  (151.)* 
and  (152.),  equals  90  degrees,  less  the  quotient  derived  from  a  di- 
vision of  360  by  twice  the  number  of  sides  to  the  polygon. 

441. — Pentagon:  Radius  of  the  Circumscribed  and  In- 
scribed Circles:  Area.— The  rules  for  polygons  developed 
in  the  two  former  articles  will  here  be  exemplified  in  their 
application  to  the  case  of  a  regular  pentagon,  or  polygon  of 
five  sides. 

To  obtain  the  angle  c°  (153.),  we  have  n  =  5,  and — 

,•>=  90°-  3g  =  go-36  =  54°. 

For   the   radius   of  the   circumscribed   circle,  we   have 

(150.)- 


2  COS.  C 


464  POLYGONS. 

b 


2  cos.  54C 

i 
2  cos.  54° 

Using  a  table  of  logarithmic  sines  and  tangents  (Art.  427), 
we  have — 

Log.  2  =0-3010300 
Cos.  54°  =  9-7692187 

Their  sum  =  0-0702487  —  subtracted  from 
Log.  i  =  o-ooooooo 

0-85065  =9-9297513 

Therefore — 

,£  =  0-85065  £. 

Or  :  The  radius  of  the  circumscribed  circle  of  a  regular/^ta- 
gon-  equals  a  side  of  the  pentagon  multiplied  by  the  decimal  o  •  8  5065 . 

For  the  radius  of  the  inscribed  circle,  we  have  (151.)— 


=  -  tan.  ct 


.tan.  54° 
r  =  o 


For  this  we  have — 


Log.  tan.  54°  =  0-1387390 
Log.  2  =  0-3010300 

0-68819  =  9-8377090. 
Therefore — 

r  =  0-68819  b. 

Or:  The  ra&usofthe  inscribed  circle  of  a  regular  pentagon 
equals  a  side  of  the  pentagon  multiplied  by  the  decimal  0-68819. 
For  the  area  we  have  (152.)— 

A  =%fr*n  tan.  c, 
A  =  J  x  5  tan.  54°  b\ 
A  =ftan.  54°  b\ 


TABLE  FOR  REGULAR  POLYGONS.          465 

For  this  we  have — 

Log.  5.  =  0-6989700 
Log.  tan.  54°  =_-  o- 1387390 

0-8377090 
Log.  4  =  0-6020600 

1-72048  —  0-2356490 
Therefore— 

A  =  i-  72048  b  \ 

Or:  The  area  of  a  regular  pentagon  equals  the  square  of  its 
side  multiplied  by  I  •  72048. 

1-42. —  Polygons  Table  of  €on§tant  multipliers.  —  To 

obtain  expressions  for  the  radii  of  the  circumscribed  and  in- 
scribed circles,  and  for  the  area  for  polygons  of  7,  9,  10,  n, 
13,  14,  and  15  sides/a  process  would  be  needed  such  pre- 
cisely as  that  just  shown  in  the  last  article  for  a  pentagon, 
except  in  the  value  of  n  and  c,  which  are  the  only  factors 
which  require  change  for  each  individual  case. 

No  useful  purpose,  therefore,  can  be  subserved  by  ex- 
hibiting the  details  of  the  process  required  for  these  several 
polygons.  The  values  of  the  constants  required  for  the 
radii  and  for  the  areas  of  these  polygons  have  been  com- 
puted, and  the  results,  together  with  those  for  the  polygons 
treated  in  former  articles,  gathered  in  the  annexed  Table  of 
Regular  Polygons. 

REGULAR  POLYGONS. 


SIDES. 

R 

b     . 

r 
1>  = 

A 
b~*~ 

o    Trie-on 

•  ^77^ 

•28868 

•4TJOI 

A    Tetragon     

•7O7II 

•  5OOOO 

I  -OOOOO 

5    Pentagon                                 

.8co6t; 

•68819 

I  •  72O48 

•ooooo 

.  86603 

2.  cr\So8 

•152-18 

I    03826 

3-6T3QI 

8    Octagon                     

*  30656 

I  •  2O7  I  I 

4-  8284^ 

•  4.6  1  QO 

I  •37'374 

6-  18182 

10    Decagon                     ...        . 

•61803 

i  -^884 

II.   Undecagon  

•  7747^ 

i  •  70284 

<J94^J- 

9.  a6c6d. 

12    Dodecagon 

.  Q^lS^ 

i  •  86603 

13.  Tredecagon   

2  •  08020 

2-02858 

11  '  I  8^77 

14    Xetradecagon            ...     . 

2  •  24608 

2  •  I  0064 

15.   Pentadecagon  

2  •  4OJ.8  7 

2  •  ^^2^1 

XD    JJ451 
17  •  6j.2^6 

16    Hecadecagon 

2  •  56202 

2.  e  T'l67 

466  POLYGONS. 

In  this  table  R  represents  the  radius  of  the  circumscribed 
circle ;  r  the  radius  of  the  inscribed  circle ;  b  one  of  the 
sides,  and  A  the  area  of  the  polygon.  By  the  aid  of  the 
constants  of  this  table,  R,  the  radius  of  the  circumscribed 
circle  of  any  of  the  polygons  named,  may  be  found  when  a 
side  of  the  polygon  is  given.  For  this  purpose,  putting  m 
for  any  constant  of  the  table,  we  have — 

R  =  bm.  (1S4-) 

As  an  example :  let  it  be  required  to  find  R,  for  a  penta- 
gon having  each  side  equal  to  5  feet ;  then  the  above  expres- 
sion becomes — 

R  —  5  x  0-85065, 
R  =  4-25325- 

The  radius  will  be  4  feet  3  inches  and  a  small  fraction. 
In  like  manner  the  radius  of  the  inscribed  circle  will  be — 

r  =  bm;  (1550 

and  for  a  pentagon  with  sides  of  5  feet,  we  have — 

r  =  5  x  0-68819, 
r  —  3-44095- 

Or,  the  radius  of  the  inscribed  circle  will  be  3  ft.  -j^  and  a 
small  fraction.  Or,  multiplying  the  decimal  by  12,  3  ft.  5  in. 
-f^Q  and  a  small  fraction. 

The  area  of  any  polygon  of  the  table  may  be  obtained 
by  this  expression— 

A^b*m;  (156.) 

and,  applying  this  to  the  pentagon  as  before,  we  have — 

A  —  5 2  x  i  •  72048, 
A  —  43-012. 


EXPLANATION   OF  THE   TABLE.  467 

Or,  the  area  of  a  pentagon  having  its  sides  equal  to  5  feet, 
is  43  feet  and  T-^f7  of  a  foot. 

By  the  constants  of  the  table  a  side  of  any  of  its  poly- 
gons may  be  found,  when  either  of  the  radii,  or  the  area, 
are  known. 

When  R  is  known,  we  have — 


When  r  is  known,  we  have- 


When  the  area  is  known,  we  have — 


SECTION  XL— THE  CIRCLE. 


443. — Circles:  Diameter  and  Perpendicular:  mean 
Proportional. — Let  ABC  (Fig.  294)  be  a  semicircle.  From 
C,  any  point  in  the  curve,  draw  a  line  to  A  and  another  to 
B\  then  ABC  will  be  a  right-angled  triangle  (Art.  352). 
Draw  the  line  CD  perpendicular  to  the  diameter  AB\ 
then  C  D  w\\\  divide  the  triangle  A  B  C  into  two  triangles, 
A  CD  and  C B D,  which  are  homologous.  For,  let  the 
triangle  C  B  D  be  revolved  on  D  as  a  centre  until  its  line 
CD  shall  come  to  the  position  E  D,  and  the  line  DB  oc- 
cupy the  position  D  F,  each  in  a  position  at  right  angles 


to  its  former  position,  the  point  B  describing  the  curve 
B  F,  and  the  point  C  the  curve  C  E,  and  each  forming  a 
quadrant  or  angle  of  ninety  degrees.  Since  these  points 
have  revolved  ninety  degrees,  therefore  the  three  lines  of 
the  triangle  CBDhave  revolved  into  a  position  at  right 
angles  to  that  which  they  before  occupied  ;  hence  the  line 
EFis  at  right  angles  to  CB}  and  (from  the  fact  that  A  C  B  is 
a  right  angle)  parallel  with  A  C,  Since  the  triangle  EFD 
equals  the  triangle  C  B  D.  and  since  the  lines  of  E  FD  are 
parallel  respectively  to  the  corresponding  lines  of  A  CD, 
therefore  the  triangles^  CD  and  C  B  D  are  homologous. 

Comparing  the  lines  of  these  triangles  and   putting  a  = 
A  B,  y  =  C  D,  and  x  —  D  B,  we  have — 


RADIUS   FROM   CHORD  AND   VERSED   SINE.  469 

DB  :  D  C  :  :  D  C  :  A  D, 

x  :  y  :  :  y  :  a  —  x, 
y*  =  x  (a  —  x],  (160.) 

Or,  in  a  semicircle,  a  perpendicular  to  the  diameter  terminated 
by  the  diameter  and  the  curve  is  a  geometric  mean,  or  mean 
proportional,  between  the  two  parts  into  which  the  perpendicular 
divides  the  diameter. 

444. — Circle :  Radius  from  Given  Chord  and  Versed 
Sine. — Let  A  B  (Fig.  295)  be  a  given  chord  line  and  CD  a 
versed  sine.  Extend  CD  to  the  opposite  side  of  the  circle ; 
it  will  pass  through  F,  the  centre.  Join  A  and  C,  also  E  and 


B 


B.  The  line  A  D,  perpendicular  to  the  diameter  C  E,  is 
a  mean  proportional  between  the  two  parts  CD  and  DE 
(Art.  443)  ;  or,  putting  a  =  A  D,  b  =  C  D,  and  r  equal  the 
radius  FE,  we  have  — 

C  D  \  AD  \\  AD  \  DE\ 

b  :  a  :  :  a  :  2  r  —  b, 


-2  rb-b*, 
+  l>*  =  2rb, 


r  - 


(161.) 


470 


THE   CIRCLE. 


Or :  The  radius  of  a  circle  equals  the  sum  of  the  squares  of  half 
the  chord  and  the  versed  sine,  divided  by  twice  the  'versed  sine. 

Another  expression  for  the  radius  may  be  obtained ;  for 
the  two  triangles  C B  D  and  C E  B  (Fig.  295)  are  homologous 
(Art.  443)  and  their  corresponding  lines  in  proportion.  Put- 
ting/for CB,  we  have — 


or — 
or — 
and — 


CD  :  CB  :  :   CB  :   C  E, 

v  :/::/:  2  r, 

f  '  —  2  rv, 


r  = 


2V 


(162.) 


Or :  The  radius  of  a  circle  equals  the  square  of  the  chord  of  half 
the  arc  divided  by  twice  the  versed  sine. 

4-45. — Circle:  Segment  from  Ordinate§. — When  the  curve 
of  a  segment  of  a  circle  is  required  for  which  the  radius  can- 
not be  used,  either  by  reason  of  its  extreme  length,  or  be- 


FIG.  296. 

cause  the  centre  of  the  circle  is  inaccessible,  it  is  desirable 
to  obtain  the  curve  without  the  use  of  the  radius.  This  may 
be  done  by  calculating  ordinates,  a  rule  for  which  will  now 
be  developed. 

Let  DC  B  (Fig.  296)  be  a  right  angle,  and  A  DB  a  cir- 
cular arc  described  from  C  as  a  centre,  with  the  radius 
B  C=  CD  =  CP.  Draw  PM  parallel  with  DC,  and  A  G 
parallel  with  C  B.  Now,  in  the  segment  A  D  G,  we  have 
given  A  G,  its  chord,  and  D  E,  its  versed  sine,  and  it  is  re- 


RULE   FOR  ORDINATES.  471 

quired  to  find  an  expression  by  which  its  ordinates,  as  P F, 
may  be  computed.     From  Art.  416,  we  have — 

PM*=CP*-CJf*<, 
or,  putting  for  these  lines  their  usual  symbols — 


now  we  have  — 


EC=  FM, 
FM=DC—DE, 

FM  =r-b. 
Then  we  have  — 


or,  putting  t  for  P  F  and  substituting  for  PM  and  FM  their 
values  as  above,  we  have  — 

t  =  y-(r-b\ 
and  for  y,  substituting  its  value  as  above,  we  have— 


/r*  -x*  -(r-b).  (163.) 

Or:  The  ordinate  in  the  segment  equals  the  square  root  of  the 
difference  of  the  squares  of  the  radius  and  the  abscissa  minus 
the  difference  of  the  radius  and  the  versed  sine. 

For  example :  let  the  chord  A  G  (Fig.  296)  in  a  given  case 
equal  20  feet,  and  the  versed  sine,  b,  or  the  rise  D  E,  equal  4 
feet ;  and  let  the  ordinates  be  located  at  every  2  feet  along 
the  chord  line,  A  G. 

In  solving  this  problem  we  require 'first  to  find  the  radius. 
This  is  obtained  by  means  of  equation 


2b 


4/2  THE   CIRCLE. 

For  a,  half  the  chord,  we  have  10  feet  ;  for  b,  the  versed 
sine,  we  have  4  feet  ;  and,  substituting  these  values,  we 
have  — 


The  radius  equals  —  H'5 

The  versed  sine  equals  —  4-0 

(r-b}=  10-5 

The  square  of  14-5,  the  radius,  equals  210-25.     Now  we 
have,  substituting  these  values  in  equation  (163.)  — 


-~-  1210-25  —  x^  —  10-5. 

The  respective  values  of  x,  as  above  required,  are  o,  2, 
4,  6,  8  and  10  Substituting  successively  for  x  one  of  these 
values,  we  shall  have,  when — 


x—    o;  t  --  y  210-25  —  o2  — 10-5  =  4. 


x-      2  ;  /-=  |/  210-25  —  2  2  —  10- 5  =  3-8614 


4;  '=-  V  210-2$  —  4*--  10-5  =  3-4374 

#  =    6;  /  =  |/  210-25  —  62  —  10-5  =:  2-7004 

*  =    8  ;  /  =  4/  210-25  —  82  —  10-  5  =  i  •  5934 


r  I0'  *  -:  1/210-25  —  io2--  10-5  =  o-o 

Values  for  /  may  be  taken  at  points  as  numerous  as  desira- 
ble for  accuracy. 

In  ordinary  cases,  however,  they  need  not  be  nearer  than 
in  this  example. 

After  the  points  are  secured,  let  a  flexible  piece  of  wood 
be  bent  so  as  to  coincide  with  at  least  four  of  the  points  at  a 
time,  and  then  draw  the  curve  against  the  strip. 

446. — Circle  :  Relation  of  Diameter  to  Circumference. 

—In  Art.  439  it  is* shown  that  the  area  of  a  polygon  equals 
the  radius  of  the  inscribed  circle  multiplied  by  half  of  a 
side  of  the  polygon  and  by  the  number  of  the  sides ;  or, 


TO   FIND   THE   CIRCUMFERENCE.  473 

A  =  r  x  —  n  =     b  n  ;  or,  the  area  equals  half  the  radius  by  a 
2         2 

side  into  the  number  of  sides  ;  or,  half  the  radius  into  the 
periphery  of  the  polygon.  Now,  if  a  polygon  have  very 
small  sides  and  many  of  them,  its  periphery  will  approxi- 
mate the  circumference  of  the  circle  inscribed  within  it  ;  in- 
deed when  the  number  of  sides  becomes  infinite,  and  conse- 
quently infinitely  small,  the  periphery  and  circumference 
become  equal.  Consequently,  for  the  area  of  the  circle,  we 
have  — 

A  =  r--c,  (164.) 

where  c  represents  the  circumference. 

By  computing  the  area  of  a  polygon  inscribed  within  a 
given  circle,  and  that  of  one  circumscribed  about  the  circle, 
the  area  of  one  will  approximate  the  area  of  the  other  in 
proportion  as  the  number  of  the  sides  of  the  polygon  are 
increased. 

For  example  :  if  polygons  of  4  sides  be  inscribed  within 
and  circumscribed  about  a  circle,  the  radius  of  which  is  I, 
the  areas  will  be  respectively  2  and  4.  If  the  polygons  have 
1  6  sides,  the  areas  are  each  3  and  a  fraction,  the  fractions 
being  unlike;  when  they  have  128  sides  the  areas  are  each 
3  •  14  and  with  unlike  fractions  ;  when  the  sides  are  increased 
to  2048,  the  areas  each  equal  3-1415  and  unlike  fractions, 
and  when  the  sides  reach  32768  in  number  the  areas  are 
equal  each  to  3-1415926,  having  like  decimals  to  seven 
places.  The  computations  have  been  continued  to  127 
places  (Gregory's  "  Math,  for  Practical  Men  "),  but  for  all 
possible  uses  in  building  operations  seven  places  will  be  found 
to  be  sufficient.  From  this  result  we  have  the  diameter  in 
proportion  to  the  circumference  as  i  :  3-  1415926,  or  as  — 


I  :  3 
i  :  3 
1:3-  1416. 

Of  these  proportions,  that  one  may  be  used  which  will  give 


474  THE   CIRCLE. 

a  result  most  nearly  approximating  the  degree  of  accuracy 
required.  For  many  purposes  the  last  proportion  will  be 
sufficiently  near  the  truth. 

For  ordinary  purposes  the  proportion  7  :  22  is  very  use- 
ful, and  is  correct  for  two  places  of  decimals;  it  fails  in  the 
third  place. 

The  proportion  113  :  355  is  correct  to  six  places  of  deci- 
mals. 

For  the  quantity  3-1415926  putting  the  Greek  letter  n 
(called  py\  and  2  r  =  d  for  the  diameter,  we  have— 

c  —  n  d.  (165.) 

To  apply  this :  in  a  circle  of  50  feet  diameter,  what  is 
the  circumference  ? 

c  =  3-1416  x  50 
c  =  1 57-08  ft. 

If  the  more  accurate  value  of  n  be  used,  we  have — 

c  =  3-1415926  x  50, 
c  =  i 57- 07963. 

The  difference  between  the  two  results  is  0-00037,  which 
for  all  ordinary  purposes,  would  be  inappreciable. 
By  the  rule  of  7  :  22,  we  have — 

c  =  5ox-3T2-  _  157.1428571, 

an  excess  over  the  more  accurate  result  above,  of  0-0632271, 
which  is  about  £  of  an  inch. 

Bv  the  rule  of  113  :  355,  we  have — 

c  =  50  x  fff  =  157-079646. 

This  result  gives  an  excess  of  only  0-000016;  it  is  sufficiently 
near  for  any  use  required  in  building. 

From  these  results  we  have  these  rules,  namely :  To 
obtain  the  circumference  of  a  circle,  multiply  its  diameter  by 


TO   FIND   THE   AREA.  475 

22,  and  divide  the  product  by  7  ;  or,  more  accurately,  multiply 
the  diameter  ^355  and  divide  the  product  by  113;  or,  by  mul- 
tiplication only,  m ultiply  the  diameter  by  3-1416;  or,  by 
3-14159^;  or,  by  3-1415926;  according  to  the  degree  of 
accuracy  required. 

And  conversely:  To  obtain  the  diameter  from  the  cir- 
cumference, multiply  the  circumference  by  7  and  divide  the 
product  by  22  ;  or,  multiply  by  113  and  divide  by  355  ;  or,  di- 
vide the  circumference  by  3-1416;  or,  by  3-14159^;  or,  by 
3-1415926. 

4-47. — Circle :  Length  of  an  Arc. — Considering  the  cir- 
cle divided  into  360°,  the  length  of  an  arc  of  one  degree  in 
a  circle  the  diameter  of  which  is  unity  may  be  thus  found. 

The  circumference  for  360°  is  3- 14159265  ; 

3. 14159265  =  0.oo8726fi4625;. 

which  equals  an  arc  of  one  degree  in  a  circle  having  unity 
as  its  diameter;  or,  for  ordinary  use  the  decimal  0-008727 
or  0-0087^  may  be  taken  ;  or  putting  a  for  the  arc  and  g  for 
the  number  of  degrees,  we  have — 

a  =  0-00872665  dg.  (166.) 

Wherefore  :  To  obtain  the  length  of  an  arc  of  a  circle, 

multiply  the  diameter  of  the  circle  by  the  number  of  degrees  in 
the  arc,  and  by  the  decimal  0-0087^,  or,  instead  thereof,  by 
0-008727. 

4.43. — Circle:  Area. — The  area  of  a  circle  may  be  ob- 
tained in  a  manner  similar  to  that  for  the  area  of  polygons 

(Art.  439),  in   which  A—Bn\  B  —  r  — ,  or— 

A  =  %  b  n  r, 

where  b  equals  a  side  of  the  polygon  and  n  the  number  of 
sides ;  so  that  b  n  equals  the  perimeter  of  the  polygon. 

Now,  if  for  the  perimeter  of  the  polygon  there  be  sub- 


476  THE   CIRCLE. 

stituted  the  circumference  of  the  circle,  we  shall  have,  put- 
ting for  the  circumference  3- 1416  dy  or,  n  d  (Art.  446)— 

A  =  \n  dr, 

in  which  r  is  the  radius.     Since  2  r  —  d,  the  diameter,  and 
r  =  -,  we  have — 

•    d 


And  since — 

^  =  3.14159265, 

\7t  =  0-78539816, 
or— 

\  n  =  0-7854,  nearly. 
Therefore— 

(167.) 


Or:  The  area  of  a  circle  equals  the  square  of  the  diameter  mul- 
tiplied by  0-7854. 

B 


As  an  example,  the  area  of  a  circle  10  feet  in  diameter  is 
found  thus — 

IOX  IO  =  IOO. 

100x0-7854  =  78 -54  feet. 

449. — Circle:  Area  of  a  Sector. — The  area  of  A  B  CD 
(Fig.  297),  a  sector  of  a  circle,  is  proportionate  to  that  of  the 
whole  circle.  For,  as  the  circumference  of  the  whole  circle 
is  to  its  area,  so  is  the  arc  A  B  C  to  the  area  of  A  B  C  D. 


AREA  OF   SECTOR.  477 

The  circumference  of  a  circle  is  (165.)  C=  ir  d.  The  area 
of  a  circle  is  (167.)  A  =  -7854  d*.  For  the  arc  ABC  put  a, 
and  for  the  area  of  A  B  CD  put  s.  Then  we  have  from  the 
above-named  proportion— 

7t  d  : 


_ 

•J       - 


* 

Tt  d 


The  coefficient  0-7854  is  J-  (^4rA  448). 

4 

Therefore,  multiplying  the  fraction  by  4,  we  have  — 
5-  *d\ 

O     -  7     ft     » 

4  TTdf 

or—  S  =  \da  =  \ra.  (168.) 

Wherefore  :  To  obtain  the  #raz  of  a  sector  of  a  circle, 
multiply  a  quarter  of  the  diameter  by  the  length  of  the  arc. 

Thus:  let  A  D  equal  10;  also  let  A  B  C  =  a,  equal  12. 
Then  the  area  of  A  £  CD  is— 

S  =%x  lox  12, 

S  =  6o. 
The  length  of  the  arc  may  be  had  by  the  rule  in  Art.  447. 

450.  —  Circle:  Area  of  a  Segment.  —  In  the  last  article, 
A  BCD  (Fig.  297)  is  called  the  sector  of  a  circle.  Of  this 
the  portion  included  within  A  E  C  B  is  a  segment  of  a  circle. 
The  area  of  this  equals  the  area  of  the  sector  minus  the  area 
of  the  triangle  A  D  C  ;  or,  putting  M  for  the  area  of  the  seg- 
ment, S  for  the  area  of  the  sector,  and  T  for  the  area  of  the 
triangle,  then— 

M=S-  T. 

Putting  c  for  A  C  (Fig.  297)  and  h  for  D  E,  then  T  =  ~  h. 
In  the  last  article,  s  —  £  ra,  in  which  a  =  the  length  of  the 


478 


THE   CIRCLE. 


arc  ABC. 
have  — 


Substituting  this  value  of  s  in  the  above,  we 


ar  — 


Or :  When  the  length  of  the  arc  is  known,  also  that  of  the 
chord  and  the  perpendicular  from  the  centre  of  the  circle, 
then  the  area  of  the  segment  equals  the  difference  between  the 
product  of  half  the  arc  into  the  radius,  aud  half  the  chord  into 
its  perpendicular  to  the  centre  of  the  circle. 

But  ordinarily  the  length  of  the  arc  and  of  the  chord  are 
unknown.  If  in  this  case  the  number  of  degrees  contained 
between  the  two  radii,  DA,DC>wz  known,  then  the  area  of 


the  segment  may  be  found  by  a  rule  which  will  now  be  de- 
veloped. 

In  Fig.  298  (a  repetition  of  Fig.  297)  upon  D  as  a  centre, 
and  with  D  F  =  unity  for  a  radius,  describe  the  arc  H F. 
Then  GFis  the  sine  of  the  angle  C  D  B,  and  D  G  is  the  co- 
sine ;  and  we  have — 


or — 


Again — 


or  — 


DF  :  GF  :  :  DC  :  EC, 


I   :  sin  :  :  r  :  -  =  r  sin. 


DF  :  DG  :  :  DC  :  D  Ey 
i   :  cos  :  :  r  :  //  =  r  cos. 


RULE  FOR  AREA  OF  SEGMENT.  479 

By  equation  (166.)  we  have — 

a  =  0-00872665  dg, 

in  which  a  is  the  length  of  the  arc  ;  g  the  number  of  degrees 
contained  in  the  arc  ;  and  d  is  the  diameter  of  the  circle. 
Since  d  =  2  r,  therefore — 

a  =  0-0174533  rg. 

Putting  B  for  the  decimal  coefficient,  we  have — 

a  =  Br  g. 

The  expression  (169.),  by  substitution  of  values  as  above, 
becomes — 

a           c 
M  =  -r h. 

2  2 

B  rg 
M  = —  r  —  r  sin.  x  r  cos. 

M  —  \  B gr*  —  sin.  cos.  r2 

M  =  r*  (%*Bg  —  sin.  cos.) 

M  —  r  *  (o  •  00872665  g  —  sin.  cos.)          ( 1 70.) 

Or :  The  area  of  a  segment  of  a  circle  equals  the  square  of  the 
radius  into  the  difference  between  0-00872665  times  the  number 
of  degrees  contained  in  the  arc  of  the  circle,  and  the  product  of 
the  sine  and  cosine  of  half  the  arc. 

When  the  number  of  degrees  subtended  by  the  arc  is 
unknown,  or  tables  of  sines  and  cosines  are  not  accessible, 
then  the  area  may  be  obtained  by  equation  (169.),  provided 
the  chord  and  versed  sine  are  known  ;  but  before  this  equa- 
tion can  be  used, for  this  purpose,  expressions  giving  their 
values  in  terms  of  the  chord  and  versed  sine  must  be  ob- 
tained, for  a,  the  arc,  r,  the  radius,  and  h,  the  perpendicular 
to  the  chord  from  the  centre  of  the  circle. 

For  the  value  of  the  arc  we  have  (from  "  Penny  Cycl.," 
Art.  Segment]  as  a  close  approximation — 


480  THE   CIRCLE. 

By  equation  (162.)  we  have  — 

=  2^' 

Then— 

h  =  r  —  v, 
or  — 

h  =  f--v. 

2  V    • 

Substituting  these  values  in  equation  (169.)  we  have  — 


This  rule  is  the  rule  (169.)  expanded. 

The  written  rule  for  equation  (169.)  may  be  used,  substi- 
tuting for  "  half  the  arc"  one  sixth  of  the  difference  between 
eight  times  the  chord  of  half  the  arc  and  the  chord  (or  \  of  8 
times  A  £>,  Fig.  298,  minus  A  C,  the  chord).  Also  substitute 
for  "  the  radius"  the  square  of  the  chord  of  half  the  arc  divided 
by  twice  the  versed  sine.  Also,  tor.  "its  perpendicular  to  the 
centre  of  the  circle"  substitute,  the  quotient  of  the  square  of  the 
chord  of  half  the  arc  divided  by  twice  the  versed  sine,  minus 
the  versed  sine. 

When  the  arc  is  small  the  curve  approximates  that  of  a 
parabola.  In  this  case  the  equation  for  the  area  of  the  par- 
abola, which  is  quite  simple,  may  be  used.  It  is  this  — 


Or,  in  segments  of  circles  where  the  versed  sine  is  small  in 
comparison  with  the  chord,  the  area  equals  approximately  two 
thirds  of  the  chord  into  the  versed  sine. 


SECTION   XII.— THE   ELLIPSE. 

451.— Ellipse  :  Definitions.— Let  two  lines,  PF,  PF'  (Fig. 
299),  be  drawn  from  any  point  P  to  any  two  fixed  points 
FF'y  and  let  the  point  P  move  in  such  a  manner  that  the  sum 
of  the  two  lines,  PF,  PF',  shall  remain  a  constant  quantity ; 
then  the  curve  P M  KO  G  A  D  B  P,  traced  by  P,  will  be  an 
Ellipse  ;  the  two  fixed  points  F,  F' ,  the  Foci ;  the  point  C  at 


FIG.  299. 

the  middle  of  FF',  the  centre  ;  the  line  A  M  drawn  through 
F  F'  and  terminated  by  the  curve,  the  Major  or  Transverse 
Axis  ;  the  line  B  O,  drawn  through  C  and  at  right  angles  to 
A  M,  the  Minor  or  Conjugate  Axis;  the  line  G  P,  drawn 
through  Pand  C  and  terminated  by  the  curve,  the  Diameter 
to  the  point  P;  the  line  D K  drawn  through  C,  parallel  with 
the  tangent  P  T,  and  terminated  by  the  curve,  the  diameter 
Conjugate  to  P  G\  the  line  EH R  drawn  parallel  with  D  K 
is  a  double  ordinate  to  the  abscissas  G H "and  H Poi  the  di- 
ameter GP(EH  =  HR)  ;  the  line  JL  drawn  through  Fat  a 


482 


THE   ELLIPSE. 


right  angle  to  A  M  and  terminated  by  the  curve,  the  Param- 
eter, or  Latus  Rectum. 

When  the  point  P  reaches  and  coincides  with  B,  the  two 
lines  PFand  PF'  become  equal. 

The  proportion  between  the  major  and  minor  axes  de- 
pends upon  the  relative  position  of  F,Ff,  the  foci ;  the  nearer 
these  are  placed  to  the  extremities  of  the  major  axis  the 
smaller  will  the  minor  axis  be  in  comparison  with  the  major 
axis.  The  nearer  F,  F'  approach  C,  the  centre,  the  nearer 
will  the  minor  axis  approach  the  length  of  the  major  axis. 
When  F,  F'  reach  and  coincide  with  the  centre,  the  minor 
axis  will  equal  the  major  axis,  and  the  ellipse  will  become  a 


circle.  Then  we  have  PF  =  PF'  =  B  C=  A  C.  From  this 
we  \earnPF+PF'=2A  C=AM-t  also,  when  PF=  PF', 
thenPF=£F=AC. 

From  this  we  may,  with  given  major  and  minor  axes, 
find  the  position  of  F  and  Ff .  To  do  this,  on  B,  as  a  centre, 
with  A  C  for  radius,  mark  the  major  axis  at  F  and  F' . 

452. — Ellip§e  :  Equations  to  the  Curve. — An  equation  to 
a  curve  is  an  expression  containing  factors  two  of  which, 
called  co-ordinates,  measure  the  distance  to  any  point  in  the 
curve.  For  example  :  in  a  circle  it  has  been  shown  (Art. 
443)  that  P  N  is  a  mean  proportional  to  A  A^and  N B.  Or, 
putting  x  —  A  N,  y  =  PN,  and  a  —  A  B,  we  have — 

AN  :  PN  :  :  PN  :  NBy 


or — 
or — 


x  :  y  :  :  y  :  a  —  x, 

y  a  —  X  (a  —  X}. 


EQUATION   TO   THE   ELLIPSE.  483 

This  is  the  equation  to  the  circle  having  the  origin  of  x 
and  y,  the  co-ordinates  at  A,  the  vertex  of  the  curve.  It  will 
be  observed  that  the  factors  are  of  such  nature  in  this  equa- 
tion, that  it  may  be  employed  to  measure  the  distance,  rect- 
angularly, to  (*,  wherever  in  the  curve  the  point  P  may  be 
located.  By  this  equation  the  rectangular  distance  to  any 
and  every  point  in  the  curve  may  be  measured  ;  or,  having 
the  curve  and  one  of  the  lines  ;ror  y,  the  other  may  be  com- 
puted. 

From  this  example,  the  nature  and  utility  of  an  equation 
to  any  curve  may  be  understood.  The  equation  to  the 
ellipse  having  the  origin  of  co-ordinates  at  the  vertex,  is 
similar  to  that  for  the  circle.  In  the  form  usually  given  by 
writers  on  Conic  Sections,  it  is — 


in  which  a  —  'A  C  (Fig.  299)  ;  b  =  B  C\  x  equals  A  N,  and  y  = 
PN. 

If,  as  before  suggested,  the  loci  be  drawn  towards  the  cen- 
tre and  finally  made  to  coincide  with  it,  the  minor  axis  would 
then  become  equal  to  the  major  axis,  changing  the  ellipse  into 
a  circle.  In  this  case,  the  factors  a  and  b  in  the  equation  would 

become  equal;  and  the  fraction  —5-  would  equal  —  ,  =  i,and 

a  a 

hence  the  equation  would  become  — 


or— 


y  a  =  x  (2  a  —  x)  ; 


precisely  the  same  as  in  the  equation  to  the  circle  above 
shown.  The  2  a  of  this  equation  is  equivalent  to  a  of  the 
circle  ;  for  a  in  the  ellipse  represents  only  half  the  major 
axis  ;  while  in  the  equation  to  the  circle  a  represents  the 
diameter.  The  relation  between  the  ellipse  and  the  circle 
is  thus  shown  ;  indeed,  the  circle  has  been  said  to  be  an 
ellipse  in  its  extreme  conditions. 


484 


THE  ELLIPSE. 


453.—  Ellipse  :  Relation  of  Axi§  to  Abscissas  of  Axe§ 

Multiplying  equation  (173.)  by  a*  we  have— 

a*  y*  =  b*  (2  ax-x*\ 
or—  #2j/2  =  b*x(2  a  —  x).          ^ 

These  four  factors  may  be  put  in  a  proportion,  thus  — 


rfa  :  b*  :  :  x  (2  a  —  x)  :  y\ 


representing  — 


A~C* 


7TC 


NX  NM  :  P  N\ 


Or  :  The  rectangle  of  the  two  parts  into  which  the  ordinate 
divides  the  axis  major  is  in  proportion  to  the  square  of  the 
ordtnate,  as  the  square  of  the  semi-axis  major  is  to  the  square 
of  the  semi-axis  minor. 


It  is  shown  by  writers  on  Conic  Sections  that  this  rela- 
tion is  found  to  subsist,  not  only  with  the  axes  and  ordinate, 
but  also  between  an  ordinate  to  any  diameter  and  the  ab- 
scissas of  that  diameter  ;  for  example,  referring  to  Fig.  299  — 


If  A  B'  P'M  (Fig.  301)  be  a  semi-circle,  then  (Art.  443) 
*  =  A 


Substituting  this  value  of  A  NX  N Min — 

TC*  :  ~B~C*  -  v^Wa  :  ~PN*> 
we  have — 

A  C  i  BC  :  :  P'N  :  PN\ 


RELATION   OF  TANGENT  TO   AXIS.  485 

Or  :  The  ordinate  in  the  circle  is  in  proportion  to  its  correspond- 
ing ordinate  in  the  ellipse,  as  the  semi-axis  major  is  to  the  semi- 
axis  minor,  or  as  the  axis  major  is  to  the  axis  minor. 

454. — Ellipse  :  Relation  of  Parameter  and  Axe§. — The 

equation  to  the  ellipse  when  the  origin  of  the  co-ordinates 
is  at  the  centre  is,  as  shown  by  writers  on  Conic  Sections, 
thus — 

a* y*  =  a*b*-b*  x'\  (174.) 

or —  a* y"1  —  b*  (a*  —  x' 2). 

If  x'  equal  C-F  (Fig.  299)  then  the  ordinate  will  be  located 

^-       '^^ 

01    THE  ^f 

.'UNIVERSITY 

Then- 


This  is  shown  also  by  the  figure. 

Substituting  in  the  above  this  value  of  a*  —  x'*,  we  have  — 

a*y*  =  &*&*  =  b\ 
From  which,  taking  the  square  root— 

ay  =  b\ 

or  —  a  :  b  :  :  b  :  y. 

Now  y,  located  at  FJ,  is  the  semi-parameter;  hence  we 
have  the  semi-minor  axis  a  third  proportional  to  the  semi- 
major  axis  and  the  semi-parameter.  Or  :  The  parameter  is  a 
third  proportional  to  the  two  axes  of  an  ellipse. 

455.  —  Cllip§e:  Relation  of  Tangent  to  the  Axes.  —  Let 

T  T'  (Fig.  301)  be  a  tangent  to  P,  a  point  in  the  ellipse  ;  then, 
as  has  been  shown  by  writers  on  Conic  Sections  — 


or— 


CM  :  CT  ::  CN  :  CM. 


486  THE    ELLIPSE. 

Or  :  The  semi-major  axis  is  a  mean  proportional  between  the  ab- 
scissa C  N  and  C  T,  the  part  of  the  axis  intercepted  between  tJie 
centre  and  the  tangent. 

This  relation  is  found  also  to  subsist  between  the  similar 
parts  of  the  minor  axis  ;  for  — 


This  relation  affords  an  easy  rule  for  finding  the  point  T, 
or  T'  ;  for  from  the  above  we  have  — 


-    CN' 
or,  putting  /  for  C  T,  we  have — 

:?y;y  /  =  £  «75.) 

or — 

t'  =  --.  (176.) 

y 

Since  the  value  of  t  is  not  dependent  upon  y  nor  upon  b, 
therefore  /  is  constant  for  all  ellipses  which  may  be  de- 
scribed upon  the  same  major  axis  A  M\  and  since  the  circle 
is  an  ellipse  (Art.  452)  with  equal  major  and  minor  axes, 
therefore  rule  (175.)  is  applicable  also  to  a  circle,  as  shown 
in  Fig.  301. 

The  equation  (175.)  gives  the  value  of  /  =  C  T.     From 
this  deducting  CN  =  x' ,  we  have  N  T,  the  sub  tangent,  or— 

CT-  CN  =  NT, 

t  -  X>   =   S  ; 

or,  substituting  for  t  its  value  in  (175.),  we  have — 


Or:  The  subtangent  to  an  ellipse  equals  the  difference  between 
the  quotient  of  the  square  of  the  semi-major  axis  divided  by  tlie 
abscissa,  and  the  abscissa  ;  the  origin  of  the  co-ordinates  being 
at  C,  the  centre. 


AXES  TO  CONJUGATE  DIAMETER. 


487 


456. — Ellipse  :  Relation  of  Tangent  witli  the  Foci. — Let 

the  two  lines  from  the  foci  to  P  (Fig.  302),  any  point  in  the 
ellipse,  be  extended  beyond  P.     With  the  radius  P F'  de- 


FIG.  302. 

scribe  from  P  the  arc  F'  G,  and  bisect  it  in  H.     Then  the 
line  P  T,  drawn  through  H,  will  be  a  tangent  to  the  ellipse 

Sit  P. 

This  has  been  shown  by  writers  on  Conic  Sections.  The 
construction  here  shown  affords  a  ready  method  of  drawing 
a  tangent.  And  from  the  principle  here  given  we  learn 
that  a  tangent  makes  equal  angles  with  the  lines  from  the 
tangential  point  to  the  two  foci. 

For,  because  GH=  HF',  we  have  the  angle  F'  PH  = 
HPG.  The  angles  H  PG  and  KPF  are  opposite,  and 
hence  (Art.  344)  are  equal ;  and  since  the  two  triangles 
F'PffandKPFare  each  equal  to  HPG,  therefore  F' PH 
and  KPF  are  equal  to  each  other.  Or:  A  tangent  to  an 
ellipse  makes  equal  angles  with  the  tivo  lines  drawn  from  the 
point  of  tangency  to  the  two  foci. 

Experience  shows  that  light  shining  from  one  focus  is 
reflected  from  the  ellipse  into  the  other  focus.  It  is  for  this 
reason  that  the  two  points  F  and  F'  are  called  foci,  the  plu- 
ral oifoczts,  a  fireplace. 

457.— Ellipse :  Relation  of  Axes  to  Conjugate  Diame- 
ter—Parallel with  K  T  (Fig.  302)  let  D  E  be  drawn  through 


488  THE   ELLIPSE. 

C,  the  centre,  and  L  Q  through  y,  one  end  of  the  diameter 
from  the  point  P.  Parallel  with  this  diameter  PJ  draw  L  K 
and  QR  through  the  extremities  of  the  diameter  D  E.  Then 
D  E  is  a  diameter  conjugate  to  the  diameter  PJ,  and  K  R, 
R  Q,  QL,  and  L  K  are  tangents  at  the  extremities  of  these 
conjugate  diameters. 

Now  it  is  shown  by  writers  on  Conic  Sections  (Fig.  302) 
that— 


PC*, 


or  — 


Or  :  The  sum  of  the  squares  of  the  two  axes  equals  the  sum 
of  the  squares  of  any  two  conjugate  diameters. 

From  this  it  is  also  shown  that  the  area  of  the  parallelo- 
gram K  C  equals  the  rectangle  A  C  x  B  C',  or,  that  a  paral- 
lelogram formed  by  tangents  at  the  extremities  of  any  two 
conjugate  diameters  is  equal  to  the  rectangle  of  the  axes. 

458.  —  Ellipse  ;  Area.—  Let  E  equal  the  area  of  an  ellipse  ; 
A  the  area  of  a  circle,  of  which  the  radius  a  equals  the  semi- 
major  axis  of  the  ellipse,  and  let  b  equal  the  semi-minor  axis. 
Then  it  has  been  shown  that  — 

E  :  A   :  :  b  :  a, 

E=Ab-. 
a 

The  area  of  a  circle  (Art.  448)  is  — 

A  —  \  n  dr  =  TT  r*, 
and  when  the  radius  equals  a  — 

A  =  n  a  2, 
This  value  of  A,  substituted  in  the  above  equation,  gives— 

E   =    TTtf2-, 

a 

E  =  n  ab.  (178.) 


PRACTICAL  SUGGESTIONS. 


489 


Or:   The  area  of  an  ellipse  equals  3- 141 59^  times    the  product 
of  the  semi-axes  ;  or  0-7854  times  the  product  of  the  axes. 

459. — Ellipse  :  Practical  Suggestion*. — In  order  to  de- 
scribe the  curve  of  an  ellipse,  it  is  essential  to  have  the  two 
axes  ;  or,  the  major  axis  and  the  parameter  ;  or,  the  major 
axis  and  the  focal  distance. 

If  the  two  axes  are  given,  then  with  the  semi-major  axis 
for  radius,  from  B  (Fig.  299)  as  centre  an  arc  may  be  made 
at  F  and  F't  the  foci ;  and  then  the  curve  may  be  described 
by  any  of  the  various  methods  given  at  Arts.  548  to  552. 

If  the  major  axis  only  and  the  parameter  are  given,  then 
(Art.  454)  since — 

*»  =  ay, 
we  have — 


=    Vay. 


(I79-) 


Or :  The  semi-minor  axis  of  an  ellipse  equals  the  square  root 
of  the  product  of  the  semi-major  axis  into  the  semi-parameter. 
Then,  having  both  of  the  axes,  proceed  as  before. 

If  the  major  axis  and  the  focal  distance  are  given,  or  the 
location  of  the  foci ;  then  with  the  semi-major  axis  for   ra- 


N    N 

FIG.  303. 

dius  and  from  the  focal  points  as  centres,  describe  arcs  cut- 
ting each  other  at  B  and  O  (Fig.  299).  The  intersection  of 
the  arcs  gives  the  limit  to  B  O,  the  minor  axis.  With  the 
two  axes  proceed  as  before.  Points  in  the  curve  may  be 
found  by  computing  the  length  of  the  ordinates,  and  then 
the  curve  drawn  by  the  side  of  a  flexible  rod  bent  to  coin- 
cide with  the  several  points.. 

For   example,  let  it  be  required  to  find  points  in  the 
curve  of  an  ellipse,  the  axes  of  which  are  12  and  20  feet ;  or 


490  THE   ELLIPSE. 

the  semi-axes  6  and  10  feet,  or  6  x  12  =  72  inches,  and  10  x 
12  =  1  20  inches. 

Fix  the  positions  of  the  points  N  N',  etc.,  along  the  semi- 
major  axis  C  '  M  (Fig.  303)  at  any  distances  apart  desirable. 
It  is  better  to  so  place  them  that  the  ordinates  when  drawn 
shall  divide  the  curve  B  PM'mto  parts  approximately  equal. 
If  CM  be  divided  into  eight  parts  as  shown,  these  parts 
measured  from  C  will  be  well  graded  if  made  equal  severally 
to  the  following  decimals  multiplied  by  CM.  In  this  case 
CM=  120;  therefore  — 

C  N   —  1  20  x  0-3       =  36-  =  x' 

C  N'  —  120  x  0-475  =  57-  =  x' 

C  Nff  =  120  x  0-625  =  75  -  =  x' 

Etc.,  =  120x0-75     =  90-  —  xf 

120  X  0-85      =    102-    =  Xf 

120  x  0-925  =  in  -  =  x' 
120x0-975  =  117-  =•  _x' 

1  2O  X    I-O          =    1  2O-    —   X'  . 

The  equation  of  the  ellipse  having  the  origin  of  co-ordi- 
nates at  the  centre  (Art.  454)  is  — 


or,  dividing  by  a*- 


a 


or — 


or-  y=-\/  a*~xr*;  (i  go.) 

in  which  a  and  b  represent  the  semi-axes.     Substituting  for 
these  their  values  in  this  case,  we  have — 


LENGTH   OF  ORDINATES.  491 

Now,  substituting  in  this  equation  the  several  values  of 
x*  successively,  the  values  of  the  corresponding  ordinates 
will  be  obtained.  For  example,  taking  36,  the  first  value  of 
x',  as  above,  we  have — 

y  =    0-6  ^74400  —  36* 
y  =  68-684; 


y  =.    0-6  V 14400  —  57 2 

y  =  63-359; 

and  so  in  like  manner  compute  the  others. 

The  ordinates  for  this  case  are  as  follows,  viz. : 

When  x'  —      o,  y  —  72-0 
"      x'  —     36,  y  —  68-684 
"      £n=     57,^  =  63.359 

"    x'  =    75,  y  =  56-205 
"    *  =  90,  y  =  47-624 

"       X'  =    102,  J/  =   37-928 

"      x'  =  ill,  7  =  27-358 

"     /=;  117,  y  =  15-999 

"       *'  =    120,  J/  —      0-0. 

The  computation  of  these  ordinates  is  accomplished  easv 
ly  by  the  help  of  a  table  of  square  roots  and  of  logarithms. 

For  example,  the  work  for  one  ordinate  is  all  comprised 
within  the  following,  viz. : 


y  —  0-6  1/14400  —  362  =  68-684. 

I2O2  =  I44OO 
36  2  =   I2Q6 

I3I04  =  4-JI74Q39 

Half    =  2-0587020 

0-6  =  9-7781513 

68-684  —  1-8368533. 

The  logarithm  of  13104  =  4-1 174039.  The  half  of  this  is 
the  logarithm  of  the  square  root  of  13104.  To  the  half  log- 
arithm add  the  logarithm  of  c-6;  the  sum  is  the  logarithm 
of  68-684  found  in  the  table  (see  Art.  427). 


SECTION   XIII.— THE  PARABOLA. 

460. — Parabola  :  Definitions. — The  parabola  is  one  of 
the  most  interesting  of  the  curves  derived  from  the  sections 
of  a  cone.  The  several  curves  thus  produced  are  as  fol- 
lows :  When  cut  parallel  with  its  base  the  outline  is  a  circle  ; 
when  the  plane  passes  obliquely  through  the  cone,  it  is  an 
ellipse  ;  when  the  plane  is  parallel  with  the  axis,  but  not  in 
the  axis,  it  is  a  hyperbola  ;  while  that  which  is  produced  by 


FIG.  304. 


a  plane  cutting  it  parallel  with  one  side  of  the  cone  is  a 
parabola. 

Let  the  lines  L  M  and  L  N  (Fig.  304)  be  at  right  angles  ; 
draw  CFB  parallel  with  L M\  make  LQ  —  LF\  draw  QB 
parallel  with  LF\  then  FB  =  B  Q.  Now  let  the  line  A  L 
move  from  F L,  but  remain  parallel  with  it,  and  as  it  moves 
let  it  gradually  increase  in  length  in  such  manner  that  the 
point  A  shall  constantly  be  equally  distant  from  the  line  LM 
and  from  the  point  F.  Then  A  BP,  the  curve  described  by 
the  point  A,  will  be  a  semi-parabola.  For  example,  the 
lines  FB  and  B  Q  are  equal ;  the  lines  PPand  PJ/are  equal, 
and  so  of  lines  similarly  drawn  from  any  point  in  the  curve 
A  B  P.  Let  PNbe  drawn  parallel  with  LM ;  then  for  the 


EQUATION  TO   THE  CURVE.  493 

point  P,  A  Nis  the  abscissa  and  N P  its  ordinate  (see  Art. 
452). 

The  double  ordinate  C B  drawn  through  F,  the  focus,  is 
the  parameter.  A  F  is  the  focal  distance.  A  is  the  vertex  of 
the  curve.  The  line  L  M  is  the  directrix. 

4-6 1. — Parabola  :  Equation  to  the  Curve. — In  Fig.  304 
FPN  is  a  right-angled  triangle,  therefore — 


=  FP*  - 

but-  FP  =  .MP=LN=AN+AL\ 

and—  FN=  A  N—  A  F. 

Therefore — 


NP*  =  A  N+AL*-  A  N-  A  F*  ; 
or-  '  =   * 


/  being  put  for  the  distance  LF=  FB  (see  Art.  452).     As 
in  Arts.  412  and  413,  we  have  — 


y*  =  2px  (181.) 

by  subtraction.  This  is  the  usual  equation  to  the  parabola, 
in  which  we  have  the  rule  :  The  square  of  the  ordinate  equals 
the  rectangle  of  the  corresponding  abscissa  with  the  param- 
eter. 

From  (181.)  we  have  — 

x  :  y  :  :  y  :  2p, 

or:  1\\e  parameter  is  a  third  proportional  \.o  the  abscissa  and  its 
corresponding  ordinate. 

462.  —  Parabola  :  Tangent.  —  From  M,  any  point  in  the 
directrix,  draw  a  line  to  Ft  the  focus  (Fig.  305)  ;  bisect  M  F 
in  R,  and  through  R  draw  U  T  perpendicular  to  MF,  then 
the  line  T  U  will  be  a  tangent  to  the  curve.  For,  draw  M  D 


494 


THE   PARABOLA. 


perpendicular  to  L  V,  and  from  P,  the  point  of  its  intersection 
with  the  line  TU,  draw  a  line  to  F,  the  focus ;  then,  because 
fiPis  a  perpendicular  from  the  middle  of  MF,  MPFis  an 
isosceles  triangle,  and  therefore  the  lines  MPand  FP  are 
equal,  or  the  point  P  is  equidistant  from  the  focus  and  from 
the  directrix,  and  therefore  is  a  point  in  the  curve. 

To  show  that  the  line  TU  touches  the  curve  but  does  not 
pass  through  it,  take  £7,  any  point  in  the  line  T  U,  other  than 


FIG.  305. 

the  point  P;  join  [7  to  Mand  to  F.  Then,  since  U  is  a  point 
in  the  line  T  U,  M  U  F,  for  reasons  above  given,  is  an  isosce- 
les triangle  ;  from  [/draw  U  F  perpendicular  to  L  V.  Now, 
if  the  point  £/be  also  in  the  curve,  the  lines  Wand  U  F, 
by  the  law  of  the  curve,  must  be  equal ;  but  U  F,  as  before 
shown,  is  equal  to  U  M,  a  line  evidently  longer  than  UV\ 
therefore,  it  is  evident  that  the  point  U  is  riot  in  the  curve. 
A  similar  absurd  result  will  be  reached  if  any  other  point 
than  the  point  U  in  the  line  U  T  be  assigned,  excepting  the 


RULE   FOR  THE  TANGENT.  495 

point  P.  Therefore  the  line  T  P  touches  the  curve  in  only 
one  point,  P  ;  hence  it  is  a  tangent. 

Parallel  with  L  V,  from  A  ,  draw  A  S,  the  vertical  tangent. 
Now  A  S  bisects  M  F  or  intersects  it  in  the  point  R.  For 
the  two  right-angled  triangles  FL  J/and  FA  R  are  homolo- 
gous ;  and  because  FA  =  A  Z,  by  construction,  therefore 
FR  =  RM. 

Or  :  The  vertical  tangent  bisects  all  lines  which  can  be  drawn 
from  the  focus  to  the  directrix. 

The  lines  PFand  FT  are  equal  ;  for  the  lines  MPand 
N  T  being  parallel,  therefore  the  alternate  angles  MPT 
and  N  TPare  equal  (Art.  345)  ;  and  because  the  line  P  T  bi- 
sects M  F,  the  base  of  an  isosceles  triangle,  therefore  the 
angles  M  P  T  and  FPTafe  equal.  We  thus  have  the  two 
angles  N  TP  and  FP  T  each  equal  to  the  angle  MPT-, 
therefore  the  two  angles  N  TP  and  FP  T  are  equal  to  each 
other  ;  hence  the  triangle  P  F  T  is  an  isosceles  triangle,  hav- 
ing the  points  T  and  P  equidistant  from  F,  the  focus. 

Also  because  the  line  MFis  perpendicular  to  P  T,  there- 
fore the  line  M  F  bisects  the  tangent  PT  in  the  point  R. 
And  because  TR  =  R  P,  therefore,  comparing  triangles 
TRFund  TPO,  TF=  F  O. 

The  opposite  angles  MPT  and  U  PD  made  by  the  two  in- 
tersecting lines  U  T  and  M  D  (Art.  344)  are  equal,  and  since 
the  angles  M  P  T  and  F  P  T  are  equal,  as  before  shown, 
therefore  the  angles  FP  T  and  U  PD  are  equal. 

It  is  because  these  two  angles  are  equal,  that,  in  reflectors, 
rays  of  light  and  heat  proceeding  from  /%  the  focus,  are  re- 
flected from  the  parabolic  surface  in  lines  parallel  with  the 
axis. 

For  an  equation  expressing  the  value  of  the  tangent,  we 
have  — 


Or  :  The  tangent  to  a  parabola  equals  the  square  root  of  tJie 
sum  of  four  times  the  square  of  the  abscissa  added  to  the  square 
of  the  ordinate. 


496  THE   PARABOLA. 

463.  —  Parabola:  Subtangeiit.  —  The  line  T  N  (Fig.  305), 
the  portion  of  the  axis  intercepted  between  T,  the  point  of 
intersection  of  the  tangent,  and  N,  the  foot  of  a  perpendicu- 
lar to  the  axis  from  P,  the  point  of  contact,  is  the  subtangent. 
The  subtangent  is  bisected  by  the  vertex,  or  TA  =  A  N. 
For,  the  two  triangles  TRA  and  TPNare  homologous; 
and,  as  shown  in  the  last  article,  the  line  M  F  bisects  PTin 
R-,  or  TR  =  R  P. 

Therefore,  we  have  — 


TR  :    TA   :  :    TP 

TR  x  TN  =  TA  x  TP, 
but—  TR=±TP; 

therefore-  J-  TP  x  TN=  TA  x  TP, 

|  TN  =  TA. 

Or  :  The  subtangent  of  z.  parabola  is  bisected  by  the  vertex  ;  or 
is  equal  to  twice  the  abscissa. 

And  because  of  the  similarity  of  the  two  triangles  TRA 
and  TP  N,  as  above  shown,  we  have  — 

NP=  2AR, 
y  =  2  A  R. 
Or  :  The  ordinate  equals  twice  the  vertical  tangent. 

464.  —  Parabola:  Normal  and  Subnormal.  —  The  line 
PO  (Fig.  305)  perpendicular  to  P  T,  is  the  normal  and  NO, 
the  part  of  the  axis  intercepted  between  the  normal  and  the 
ordinate,  is  the  subnormal.  For  the  normal,  from  similar 
triangles,  we  have  — 

TN  :  NP  :  :.  TP  :  P  O, 
TP 


2X 


DIAMETER  AND   RECTANGLE   OF  BASE.  497 

Or  :  The  normal  equals  the  rectangle  of  the  ordinate  and  tan- 
gent, divided  by  twice  the  abscissa. 

The  subnormal  equals  half  the  parameter.     For  (181.) — 


or—  NP*  =  2  FB  •  A  N. 


Dividing  by  2  A  N  gives— 

I  FB  =  ttf  (A'} 

In  the  similar  triangles  (Art.  443)  OPNand  PT  N,  we 
have — 

NO  :  NP  :  :  NP  :  NT, 

NO  =  N~P\ 
NT 

As  shown  in  the  previous  article,  N  T  —  2  A  N;  therefore — 


(B.) 

2AN 

Comparing  equations  (A.)  and  (B.),  we  have — 

NO  =  FB. 

Or  :  The  subnormal  of  a  parabola  equals  half  the  parameter, 
a  constant  quantity  for  the  subnormal  to  all  points  of  the 
curve. 

465. — Parabola:  Diameters. —  In  the  parabola  BAC 
(Fig.  306),  P D,  a  diameter  (a  line  parallel  with  the  axis)  to 
the  point  P,  is  in  proportion  to  B  D  x  D  C,  the  rectangle  of 
the  two  parts  into  which  the  base  of  the  parabola  is  divided 
by  the  diameter. 

This  may  be  shown  in  the  following  manner : 

DP=EN=EA-NA.  (A.) 


498 


THE   PARABOLA. 


For  EA  we  have,  taking  the  co-ordinates,  for  the  point 
C,  (i8i.)~ 


or- 


=  X 


or— 


EC 


=  EA. 


0       C 


For  N A  we  have,  taking  the  co-ordinates  to  the  point 
/>,  (181.)- 


or — 


=  X, 


NP'- 


or- 


(C.) 


Using  these  values  (B.)  and  (C.j  in  (A.),  we  have— 

=£A  -NA, 


'      NP*       EC*-NP* 


If  /  be  put  for  B  C  and  n  for  D  C,  then— 


TO    FIND   THE   CONSTITUENT   PARTS.  499 

and- 
then  (Art.  413)— 


ft 

or  (Art.  415)— 


2P 

=  n(l-  n) 
2P 


Dp_DCxBD 

~ 


Now,  since  2  /,  the  parameter,  is  constant,  we  have  D  P, 
the  diameter,  in  proportion  to  D  C  x  B  D,  the  two  parts  of 
the  base. 

Putting  d  for  the  diameter,  we  have  — 

(183.) 


2p 

Or  :  The  diameter  of  a  parabola  equals  the  quotient  obtained 
by  dividing  \\\Q  rectangle  —  formed  by  the  two  parts  into  which 
the  diameter  divides  the  base—  by  the  parameter. 

It  has  been  shown  by  writers  on  Conic  Sections  that  a 
diameter,  P  J  (Fig.  307),  to  any  point  Pin  a  parabola  bisects 
all  chord  lines,  SG,D£,  etc.,  drawn  parallel  with  the  tan- 
gent to  the  point/*;  the  diameter  being  parallel  with  the  axis 
of  the  parabola. 

466.  —  Parabola  :  Elements.  —  From  any  given  parabola, 
to  find  the  axis,  tangent,  directrix,  parameter  and  focus, 
draw  any  two  parallel  lines  or  chords,  SG  and  D  E  (Fig. 
307),  and  bisect  them  in  H  and  J\  through  these  points 
draw  JP\  then  JP  will  be  a  diameter  of  the  parabola  —  a 


5oo 


THE   PARABOLA. 


line  parallel  with  the  axis.  Perpendicular  to  P  J  draw  the 
double  ordinate  PQ  and  bisect  it  in  N\  through  N  and  par- 
allel with  PJ  draw  TO,  cutting  the  curve  in  A\  then  TO 
will  be  the  axis.  Make  AT- AN,  join  T  and  P\  then  TP 
will  be  the  tangent  to  the  point  P\  from  P  draw  PO  per- 
pendicular to  P  T\  then  P O  will  be  the  normal,  and  NO  the 
subnormal. 

With  NO  for  radius,  from  N  as  a  centre,  describe  the 
quadrant  O  R ;  draw  R  C  parallel  with  A  O,  cutting  the  curve 


FIG.  307. 


in  C\  from  C  draw  C B  perpendicular  to  A  O,  cutting  A  O  in 
F;  then  ^will  be  the  focus  and  C  B  the  parameter.  Make 
A  L  —  A  F;  draw  L  M  perpendicular  to  TO\  then  LM  will 
be  the  directrix.  Extend  PJ  to  meet  LM  at  M\  join  P  and  , 
F\  then,  if  the  work  has  been  properly  performed,  F  P  will 
equal  M  P. 

4-67. — Parabola  :  Described  mechanically. —  With  N  P 
(Fig.  308)  a  given  base,  and  N  A  a  given  height,  set  perpen- 


THE   CURVE   DESCRIBED    MECHANICALLY. 


501 


dicularly  to  the  base,  extend  N A  beyond  A,  and  make  A  T 
equal  to  NA  ;  join  T  and/*;  from  P  perpendicularly  to  TP 
drawP(9;  bisect  ON  in  R;  make  AL  and  A  F  each  equal 
to  N  R  ;  through  L,  perpendicular  to  L  O,  draw  D  E,  the  di- 
rectrix. 

Let  the  ruler  CDESbe  laid  to  the  line  DE,  then  with 
J  G  H,  a  set-square,  the  curve  may  be  described  in  the  fol- 
lowing manner : 

Placing  the  square  against  the  ruler  and  with  its  edge 


FIG.  308. 

J H  coincident  with  the  line  MPy  fasten  to  it  a  fine  cord  on 
the  edge  PE,  and  extend  it  from  P  to  Fy  the  focus,  and  se- 
cure it  to  a  pin  fixed  in  F.  The  cord  FP  will  equal  the 
edge  M P.  To  describe  the  curve  set  the  triangle  J  G  H  at 
M  P  E,  slide  it  gently  along  the  ruler  towards  D,  keeping  the 
edge  J  G  in  contact  with  the  ruler,  and,  as  the  square  is 
moved,  keep  the  cord  stretched  tight,  holding  for  this  pur- 
pose a  pencil,  as  at  K,  against  the  cord.  Thus  held,  as  the 
square  is  moved  the  pencil  will  describe  the  curve.  That 
this  operation  will  produce  the  true  curve  we  have  but  to 


502 


THE    PARABOLA. 


consider  that  at  all  points  the  line  FK  will  equal  KJt  which 
is  the  law  of  the  curve  (Art.  460). 

468. — Parabola  :  De§cribed  from  Point§. — With  given 
base,  N P  (Fig.  309),  and  given  height,  A  N,  to  find  the  points 
D,  F,  M,  etc.,  and  describe  the  curve.  Make  A  T  equal  to 
A  N  (Art.  462);  join  T  and  P',  perpendicular  to  TP  draw 


FIG.  309. 

PO ;  make  A  B  equal  to  twice  NO ;  take  G,  any  point  in  the 
axis  A  O,  and  bisect  B  G  in  J ;  on  y  as  a  centre  describe  the 
semi-circle  B  C G  cutting  A  L,  a  perpendicular  to  BO  in  C ; 
on  A  C  and  A  G  complete  the  rectangle  A  C DG.  Then  D  is 
a  point  in  the  curve.  Take  H,  another  point  in  the  axis ; 
bisect  B H  in.  K\  on  A"  as  a  centre  describe  the  semi-circle 
B  E  H  cutting  A  L  in  E ;  this  by  E  F  and  H  F,  gives  F,  an- 


THE   CURVE  DESCRIBED   FROM  ARCS.  503 

other  point  in  the  curve  ;  in  like  manner  procure  M,  and  as 
many  other  points  as  may  be  desired.  This  simple  and 
accurate  method  of  obtaining  points  in  the  curve  depends 
upon  two  well-established  equations  ;  one,  the  equation  to 
the  parabola,  and  the  other,  the  equation  to  the  circle.  The 
line  G  Dy  an  ordinate  in  the  parabola,  is  equal  to  A  C,  an 
ordinate  in  the  circle  B  CG;  A  G,  the  abscissa  of  the  para- 
bola, is  also  the  abscissa  of  the  circle  ;  in  which  we  have 
(Art.  443)- 

AG  :  AC  ::  AC  :  A  B, 

x  :  y  :  :  y  :  a  —  x, 


For  the  parabola,  we  have  (181.)  — 

y*=2px. 
Comparing"  these  two  equations,  we  have— 

x(a  —  x]  =  2px, 

a  —  x  =  2p, 
or  — 

EG  —  A  G=  2  p. 

By  construction  A  B  equals  2  NO,  or  twice  the  subnor- 
mal ;  the  subnormal  (Art.  464)  equals  half  the  parameter. 
Hence,  twice  the  subnormal  equals  the  parameter  —  equals 
2  p.  Therefore,  the  method  shown  in  Fig.  309  is  correct. 

469.  —  Parabola:  Described  from  Arcg.  —  Let  N  P  (Fig. 
310)  be  the  given  base  and  A  TV  the  given  height  of  the  par- 
abola. Make  A  T(Art.  462)  equal  A  N.  Join  TtoP;  draw 
PO  perpendicular  to  PT\  bisect  N  O  in  R  ;  make  A  L  and 
A  F  each  equal  to  N  R  ;  then  L  M,  drawn  perpendicular  to 
TO,  will  be  the  directrix.  Parallel  to  L  M  draw  the  lines 
B  D,  C  E,  etc.,  at  discretion.  Then  with  the  distance  B  L  for 


504 


THE   PARABOLA. 


radius,  and  on  F  as  a  centre,  mark  the  line  B  D  with  an  arc ; 
the  intersection  of  the  arc  and  the  line  will  be  a  point  in  the 
curve  (Art.  460).  Again,  with  C L  for  radius  and  on  .Fas  a 
centre,  mark  the  line  CE  with  an  arc ;  this  gives  another 
point  in  the  curve.  In  like  manner,  mark  each  horizontal 


FIG.  310. 

line  from  F  as  a  centre  by  a  radius  equal  to  the  perpendicu- 
lar distance  between  that  line  and  L  M,  the  directrix.  Then 
a  curve  traced  through  the  points  of  intersection  thus  ob- 
tained will  be  the  required  parabola. 

4-70.  —  Parabola  :  Dc§cribed  from  Ordinates.  —  With  a 
given  base,  NP(Fig.  311),  and  height,  A  N,  a  parabola  may 
be  drawn  through  points  J,  H,  G,  etc.,  which  are  the  extrem- 
ities of  the  ordinates  B  J,  C  H,  D  G,  etc.  ;  the  lengths  of  the 
ordinates  being  computed  from  the  equation  to  the  curve, 
(181.)- 


For  any  given  parabola,  in  base  and  height,  the  value  of 


THE  CURVE  FORMED  BY  ORDINATES. 


505 


/  may  be  had  by  dividing  both  members  of  the  equation  by 
2  4;;  by  which  we  have — 


y 


NP* 

2AN' 


(A.) 


from  which,  NPand  A  N  being  known,/  may  be  computed. 
With  the  value  of/,  a  constant  quantity,  determined,  the 
equation  is  rendered  practicable.     For,   taking  the  square 
root  of  each  member  of  equation  (181.),  we  have — 

y  =  V  2p  x.  (B-) 

which  by  computation  will  produce  the  value  of  y,  for  every 
assigned  value  of  x,  as  A  JS,  A  C,  A  D,  etc. 


FIG.  311. 


As  an  example :  let  it  be  required  to  compute  the  ordi- 
nates  in  a  parabola  in  which  the  base,  N ' P,  equals  8  feet,  and 
the  height,  A  N,  equals  10  feet.  With  these  values  equation 
(A.)  as  above  becomes — 


NP* 

2AN 


8"      _64 

2  X  10  ~      2O    "  ^'      * 


506  THE    PARABOLA. 

Then,  with  this  value  in  (B.)  as  above,  we  have,  for  each  or- 
dinate — 


y  '  z  '•  V  6.4*. 

In  order  to  assign  values  to  x,  let  A  N  be  divided  into 
any  number  of  parts  at  B,  C,  D,  etc.,  say,  for  convenience  in 
this  example,  in  ten  equal  parts  ;  then  each  part  will  equal 
one  foot,  and  we  shall  have  the  consecutive  values  of  x  =  i  , 
2,  3,  4,  etc.,  to  10,  and  the  corresponding  values  of  y  will  be 
as  follows.  When  — 


x  =     I,  y=   4/6.  4  x     i  = 


3= 
x  =    4,  y  =   V6-4X  "4  =   1/  25-6  =  5  -0596  = 

~32~  =  5  -6569  =  (etc.), 
38-4  =  6-  J968  = 
"^8  =  6>6933  = 


=  7-5^95  = 


With  these  values  of  y,  respectively,  set  on  the  correspond- 
ing horizontal  lines  By,  C  H,  DG,  E  S,  etc.,  points  in  the 
curve  y,  //,  G,  S,  etc.,  are  obtained,  through  which  the  curve 
may  be  drawn.  The  decimals  above  shown  are  the  decimals 
of  a  foot  ;  they  may  be  changed  to  inches  and  decimals  of  an 
inch  by  multiplying  each  by  12.  For  example:  12x0-5297 
=  6-3564  equals  6  inches  and  the  decimal  0-3564  of  an  inch, 
which  equals  nearly  |  of  an  inch. 

Near  the  top  of  the  curve,  owing  to  its  rapid  change  in 
direction  and  to  the  approximation  of  the  direction  of  the 
curve  to  a  parallel  with  the  direction  of  the  ordinates,  it  is 


THE   CURVE   FOUND    BY   DIAMETERS. 


SO/ 


desirable  to  obtain  points  in  the  curve  more  frequent  than 
those  obtained  by  dividing  the  axis  into  equal  parts. 

Instead,  therefore,  of  dividing  the  axis  into  equal  parts, 
it  is  better  to  divide  it  into  parts  made  gradually  smaller 
toward  the  apex  of  the  curve— or,  to  obtain  points  for  this 
part  of  the  curve  as  shown  in  the  following  article. 

471. — Parabola:  Described  from  Diameters. —  Let  EC 

(Fig.  312)  be  the  given  base  and  A  E  the  given  height,  placed 
perpendicularly  to  E  C.  Divide  E  C  in  several  parts  at 
pleasure,  and  from  the  points  of  division  erect  perpendicu- 
lars to  E  C.  The  problem  is  to  compute  the  length  of  these 
diameters,  as  DP,  and  thereby  obtain  points  in  the  curve,  as 
at  P.  For  this  purpose  we  have  equation  (183.),  which  gives 


B 


\ 


FIG.  312. 

the  length  of  the  diameters,  and  in  which  n  equals  D  C  (Fig. 
312),  /  equals  twice  E  C,  and  /  equals  half  the  parameter  of 
the  curve.  The  value  of  p  is  given  in  equation  (A.),  (Art, 
470),  in  which  y  equals  EC  (Fig.  312),  and  ^equals  A  E. 
Substituting  these  symbols  in  equation  (A.),  we  have— 


y>  EC'         _^ 

p~~''   2x  ~  2xAE~     2  /T' 

where  b  —  EC,  the  base,  and  h—AE,  the  height.     For 
substituting  this,  its  value,  in  equation  (183.),  we  have— 


hn.(2b  —  n) 


(184.) 


508  THE   PARABOLA. 

As  an  example  :  let  it  be  required  in  a  parabola  in  which 
the  base  equals  12  feet  and  the  height  8  feet,  to  compute  the 
length  of  several  diameters,  and  through  their  extremities 
describe  the  curve.  Then  h  will  equal  8,  and  b  12. 

If  the  base  be  divided  into  6  equal  parts,  as  in  Fig.  312, 
each  part  will  equal  2  feet.  Then  we  have  — 

h          8  8  i 


b*~  I2*~~  i44~    18  ' 
and  — 

_       h       .          N 
d—  -i  n  (2b-n), 


In  this  equation,  substituting  the  consecutive  values  of  #, 
we  have,  when— 

0x24 
n=    O,    d--     -jg-     =0 

2  X  22 


ti  —     z, 

18 

4X  20 

•M   A 

/y    _...-  -              .    A  .  A  A  A 

,    ,      6x18 

*-;    6,    ^-^g- 

8x16 
n=    8,    ^=-73—  =7-ni 


12  X  12 

n=  12,    d=  —  --  =  8-0 


The  several  diameters,  as  P  '  D,  in  Fig.  312,  may  now  be 
made  equal  respectively  to  these  computed  values  of  d,  and 
the  curve  traced  through  their  extremities. 


AREA   EQUALS   TWO   THIRDS   OF  RECTANGLE.  509 

472. — Parabola:  Area. — From  (181.),  the  equation  to 
the  parabola,  and  by  the  aid  of  the  calculus,  it  has  been 
shown  that  the  area  of  a  parabola  is  equal  to  two  thirds  of 
the  circumscribing  rectangle.  For  example :  if  the  height, 
A  E  (Fig.  312),  equals  8  feet,  and  EC,  the  base,  equals  12 
feet,  then  the  area  of  the  part  included  within  the  figure 
APC  E  A  equals  f  of  8x  12  =  1x96  =  64  feet;  or,  it  is  equal 
to  f  of  the  rectangle  A  B  C  E. 


SECTION  XIV.— TRIGONOMETRY. 

473. — Right- Angled  Triangle§:  The  Sides. — In  right- 
angled  triangles,  when  two  sides  are  given,  the  third  side 
may  be  found  by  the  relation  of  equality  which  exists  of 
the  squares  of  the  sides  (Arts.  353  and  416).  For  example, 


if  the  sides  a  and  b  (Fig.  313)  are  given,  cy  the  third  side, 
may  be  computed  from  equation  (115.)  — 


Extracting  the  square  root,  we  have  — 


When  the  hypothenuse  and  one  side  are  given,  by  transposi 
tion  of  the  factors  in  (115.),  we  have  — 


(A.) 


or — 


TWO   SIDES   GIVEN   TO   FIND   THE   THIRD.  511 

Owing  to  the  factors  being  involved  to  the  second  power  in 
this  expression,  the  labor  of  computation  is  greater  than 
that  in  a  more  simple  method,  which  will  now  be  shown. 

In  equation  (A.)  or  (B.)  the  factors  under  the  radical  may 
be  simplified.     By  equation  (i  14.)  we  have  — 


Therefore,  equation  (A.)  becomes— 


a  =  \(c  +  £)  (V  _  fy 

a  form  easy  of  solution. 

For  example  :  let  c  equal  29-732  and  b  equal  13-216,  then 
we  have  — 

29-732 

13-216 

The  sum  =  42-948 
The  difference  =  16-516 

By  the  use  of  a  table  of  logarithms  (Art.  427)  the  problem 
may  be  easily  solved  ;  thus  — 

Log.  42-948  =  1-6329429 
16-516  =  i  -2179049 

To  get  the  square  root—          2)2-  8508478 
a  —  26-6332  =  i  -4254239 

This  method  is  applicable  to  the  sides  of  a  triangle,  only  ; 
for  the  hypothenuse  it  will  not  serve.  The  length  of  the 
hypothenuse  as  well  as  that  of  either  side  may,  however,  be 
obtained  by  proportion  ;  provided  a  triangle  of  known  di- 
mensions and  with  like  angles  be  also  given. 

For  example:  in  Fig.  314,  in  which  the  two  sides  a  and 
b  are  known,  let  it  be  required  to  find  c,  the  hypothenuse. 

Draw  the  line  D  E  parallel  with  A  C,  then  the  two  trian- 
gles B  DE  and  BAG  are  homologous;  consequently  their 


512 


TRIGONOMETRY. 


corresponding  sides  are  in  proportion  (Art.  361).     Hence,  if 
d  equals  unity,  we  have — 

d  :  f  :  :  a  :  c, 
=  */,' 

from  which,  when  a  and  /  are  known,  c  is  obtained  by  sim- 
ple multiplication. 

474. —  Right- Angled  Triangle§:  Trigonometrical  Ta- 
bles— To  render  the  simple  method  last  named  available, 
the  lengths  of  d,  e  and  f  (Fig.  314)  have  been  computed  for 
triangles  of  all  possible  angles,  and  the  results  arranged  in 


FIG.  314. 

tables,  termed  Trigonometrical  Tables.  The  lines  d,  e,  and 
/,  are  known  as  sines,  cosine,  tangents,  cotangents,  etc.,  as 
shown  in  Fig.  315 — wheVe  A  B  is  the  radius  of  the  circle 
B  C H.  Draw  a  line  A  F,  from  A,  through  any  point,  C,  of  the 
arc  B  G.  From  C  draw  CD  perpendicular  to  A  B  ;  from  B 
draw  BE  perpendicular  to  A  B ;  and  from  G  draw  G  F  per- 
pendicular to  A  G. 

Then,  lor  the  angle  FA  B,  when  the  radius  A  C  equals 
unity,  CD  is  the  sine;  AD  the  cosine;  DB  the  versed  sine  ; 
BE  the  tangent;  GF  the  cotangent ;  AE  the  secant;  and 
A  Fihe  cosecant. 

But  if  the  angle  be  larger  than  one  right  angle,  yet  less 
than  two  right  angles,  as  BAH,  extend  HA  to  K  and  E  B 
to  K,  and  from  H  draw  H  J  perpendicular  to  A  J. 


TRIGONOMETRICAL  TABLES. 


513 


Then,  for  the  angle  BAH,  when  the  radius  A  H  equals 
unity,  HJ  is  the  sine  ;  A  J  the  cosine;  BJ  the  versed  sine  ; 
B  K  the  tangent ;  and  A  K  the  secant. 

When  the  number  of  degrees  contained  in  a  given  angle 
is  known,  the  value  of  the  sine,  cosine,  etc.,  corresponding  to 
that  angle,  may  be  found  in  a  table  of  Natural  Sines,  CO- 


FIG.  315. 

sines,  etc.     Or,  the  logarithms  of  the  sines,  cosines,  etc.,  may 
be  found  in  logarithmic  tables. 

In  the  absence  of  such  a  table,  and  when  the  degrees 
contained  in  the  given  angle  are  unknown,  the  values  of 
the  sine,  cosine,  etc.,  may  be  found  by  computation,  as  fol- 
lows:— Let  ABC  (Fig.  316)  be  the  given  angle.  At  any 
distance  from  B  draw  b  perpendicular  to  B  C.  By  any  scale 
of  equal  parts  obtain  the  length  of  each  of  the  three  lines  a, 
b,  c.  Then  for  the  angle  at  B  we  have,  by  proportion — 


5M- 


TRIGONOMETRY. 


c  :  b  :  :   i  -o  :  sin.      B  =  — . 

c 

c  :  a  :  :   I  «o  :  cos.     B  =  — . 

c 

a  :  b  :  :   i  -o  :  tan.     B  =  —. 

a 

b  :  a  :  :   I  -o  :  cot.     B  —  -. 


a  i  c  :  :   i  -o  :  sec.      B  =   -. 

a 

b  :  c  :  :    i  •  o   :  cosec.  B  —  --. 


Or,  in  any  right-angled  triangle,  for  the  angle  contained 
between  the  base  and  hypothenuse — 

When  perp.  divided  by  hyp.,  the  quotient  equals  the  sine. 

base         "         "    hyp.,       "  "  "     cosine. 

"       perp.       "         "    base,       "  "     tangent. 

"       base         "         "    perp.,     "  "     cotangent. 

"      hyp.         "         "    base,       "  "  "     secant. 

"       hyp.         "         "    perp.,      "  "     cosecant. 

To  designate  the  angle  to  which  a  trigonometrical  term 
applies,  the  letter  at  the  intended  angle  is  annexed  to  the 


c 
FIG.  316. 


name  of  the  trigonometrical  term  ;  thus,  in  the  above  exam- 
ple, for  the  sine  of  A  B  C  we  write  sin.  B  ;  for  the  cosine, 
cos.  B,  etc. 


EQUATIONS   TO    RIGHT-ANGLED   TRIANGLES. 


515 


By  these  proportions  the  two  acute  angles  of  a  right- 
angled  triangle  may  be  computed,  provided  two  of  the 
sides  are  known.  For  when  the  perpendicular  and  hypoth- 
enuse  are  known,  the  sine  and  cosecant  may  be  obtained. 
When  the  base  and  hypothenuse  are  known,  the  cosine  and 
secant  may  be  computed.  And  when  the  base  and  perpen- 
dicular are  known,  the  tangent  and  cotangent  may  be  com- 
puted. 

Either  one  of  these,  thus  obtained,  shows  by  the  trigo- 
nometrical tables  the  number  of  degrees  in  the  angle  ;  and, 
deducting  the  angle  thus  found  from  90°,  the  remainder  will 
be  the  angle  of  the  other  acute  angle  of  the  triangle.  For 


M      D 

FIG.  317. 

example  :  in  a  right-angled  triangle,  of  which  the  base  is  8 
feet  and  the  perpendicular  6  feet,  how  many  degrees  are 
contained  in  each  of  the  acute  angles  ? 

Having,  in  this  case,  the  base  and  perpendicular  known, 
by  referring  to  the  above  proportions  we  find  that  with 
these  two  sides  we  may  obtain  the  tangent ;  therefore— 


Referring  to  the  trigonometrical  tables,  we  find  that  0-75  is 
the  tangent  of  36°  52'  12",  nearly ;  therefore— 

The  quadrant  equals  90-  o-  o 
The  angle  B  equals  36-52-12 

The  angle  A  equals  53-07-48 


TRIGONOMETRY. 

475. — Right-Angled  Triangle§ :  Trigonometrical  Value 
of  Side§.— In  the  triangle  A  B C  (Fig.  317),  with  BP  =  i  for 
radius,  and  on  B  as  a  centre,  describe  the  arc  P D,  and  from 
its  intersection  with  the  lines  A  B  and  B  C,  draw  PM  and 
T D  perpendicular  to  the  line  B  C.  Then  from  homologous 
triangles  we  have  these  proportions  for  the  perpendicular — 

BD  :  DT  :  :  BC  :  CA, 
r  :  tan.  B  :  :  base  :  perp., 

I   :  tan.  B  :  :  a  :  b  =  a  tan.  B.  (1%S-) 

Also—  * 

-  BP  :   PM  \\BA\AC, 

r  :  sin.  B  :  :  hyp.  :  perp., 
I  :  sin.  B  :  :  c  :  b  =  c  sin.  B.  (186.) 

For  the  base,  we  have — 

BP  :  BM  :  :  BA  :  B  C, 
r  :  cos.  B  :  :  hyp.  :  base, 

I  :  cos.  B  :  :  c  :  a  =  c  cos.  B.  (l%7-) 

Again— 

TD  :  BD  ::  AC  :  B  C, 

tan.  B  :  r  :  :  perp.  :  base, 
tan.*:!::*:,^ L_.  (.88.) 

For  the  hypothenuse,  we  have — 

PM  :  PB  :  :  A  C  :  :  A  B, 
sin.  B  :  r  :  :  perp.  :  hyp., 


RULE   FOR  THE   PERPENDICULAR.  517 

sin.  B  :  i  :  :  b  :  c  =  -, .  (180.) 

sm.  B 

Again — 

BD  \BT\\BC\  BA, 

r  :  sec.  B  :  :  base  :  hyp., 

I  :  sec.  B  :  :  a  :  c  =  a  sec.  B  —  — - —  .      (IQO.) 

cos.  B 


This  substitution  of  the  cos.  for  the  sec.  is  needed  because 
tables  of  secants  are  not  always  accessible.  That  it  is  an 
equivalent  is  clear ;  for  we  have — 

BM  \  BP  :-.  BD  :  BT, 


cos.  \r\\r\  sec.  = 

cos. 


By  these  equations  either  side  of  a  right-angled  triangle 
may  be  computed,  provided  there  are  certain  parts  of  the 
triangle  given.  As,  for  example :  of  the  six  parts  of  a  tri- 
angle (the  three  sides  and  the  three  angles),  three  must  be 
given,  and  at  least  one  of  these  must  be  a  side. 

As  an  example :  let  it  be  required  to  find  two  sides  of  a 
right-angled  triangle  of  which  the  base  is  100  feet,  and  the 
acute  angle  at  the  base  is  35  degrees.  Here  we  have  given 
one  side  and  two  angles  (the  base,  acute  angle,  and  the  right 
angle)  to  find  the  other  two  sides,  the  perpendicular  and  the 
hypothenuse. 

Among  the  above  rules  we  have,  in  equation  (185.),  for 
the  perpendicular — 

b  —  a  tan.  B. 

Or :  The  perpendicular  equals  the  product  of  the  base  into  the 
tangent  of  the  acute  angle  at  the  base. 


5  1 8  TRIGONOMETRY. 

Then  (Art.  427)— 

The  logarithmic  tangent  of  B  (—  35°)  is  9-8452268 
Log.  of  a  (—  100)  is  2-0000000 

Perpendicular,  b  (=  70-02075)  =  I  -8452268 

And    for   the    hypothenuse,   taking   equation    (190.),   we 
have — 


cos.  B 

Or :    The  hypothenuse  equals  the  quotient  of  the  base  divided 
by  the  cosine  of  the  acute  angle  at  the  base. 
For  this  we  have- 
Log,  of  a  (=  100)  is  2-0000000 
"    cos. B  (=  35°)  is  9-9133645 
Hypothenuse  c(~  122-0775)  =  2-0866355 

We  thus  find  that  a  right-angled  triangle,  having  an  angle 
of  35  degrees  at  the  base,  has  its  three  sides,  the  perpendic- 


ular, baseband  hypothenuse,  respectively  equal  to  70-02075, 
loo,  and  122-0775. 

N.B. — The  angle  at  A  (Fig^ij)  is  obtained  by  deducting 
the  angle  at  B  from  90°  (Art.  346).  Thus,  90  —  35  =  55  ; 
this  is  the  angle  at  A,  in  the  above  case. 

If  the  perpendicular  be  given,  then  for  the  base  use 
equation  (188.),  and  for  the  hypothenuse  use  equation  (189.). 
If  the  hypothenuse  be  given,  then  for  the  base  use  equation 
(187.)  and  for  the  perpendicular  use  equation  (186.). 


USEFUL   RULE   FOR  THE   SIDES.  519 

476.  —  Oblique-Angled  Triangle*  :    Sine*  and  Side*.  —  In 

the  oblique-angled  triangle  A  BC  (Fig.  318)  from  C  and  per- 
pendicular to  A  B  draw  CD.  This  line  divides  the  oblique- 
angled  triangle  into  two  right-angled  triangles,  the  lines  and 
angles  of  which  may  be  treated  by  the  rules  already  given  ; 
but  there  is  a  still  more  simple  method,  as  will  now  be 
shown. 

As  shown  in  Art.  ^4:  "  When  the  perpendicular  is  di- 
vided by  the  hypothenuse  the  quotient  equals  the  sine." 
Applying  this  to  Fig.  318,  we  have  — 


jNIVERSITY 


Let  the  former  be  divided  by  the  latter  ;  then  — 

d 

sin.  A  __  b_ 
~sm7£  ~  d  ' 
a 

or,  reducing,  we  have  — 

sin.  A 


or,  putting  the  equation  in  the  form  of  a  proportion- 
sin.  B  :  sin.  A  :  :  b  :  a  ; 

or  ;  the  sines  are  in  proportion  as  the  sides,  respectively  op- 
posite. Or,  as  commonly  stated,  the  sines  are  in  proportion 
as  the  sides  which  subtend  them. 

This  is  a  rule  of  great  utility  ;  by  it  we  obtain  the  follow- 

ing : 

Referring  to  Fig.  318,  we  have- 

sin.  B  :  sin.  A   :  :  b  :  a  =  b  *-.    —  -  .  (191.) 


520  TRIGONOMETRY. 

~  sin.  A 

sin.  C  :  sin.  A  :  :  c  :  a  =  c- ~  .  (192.) 

sin.  C 

„  ,  sin.  B 

sin.  A   :  sin.  B  :  :  a  :  b  —  a  —. .  (ICH.) 

sin.  A 

~       .      „  ,          sin.  B 

sin.  C  :  sin.  B  :  :   c  :  b  =  £- -.  (iQ4.) 

sin.  (7 

,        .      ^  sin.  6"  ,        N 

sin.  A  :  sin.  c  :  :  a  :  c  =  a-^—  -~  .  (iQ5.) 

sin.  A 

•n  '         r  L  7  Sm-    C 

sin.  B  :  sin.  C  \  \  b  \  c  —  b . 

sin.  B 

These  expressions  give  the  values  of  the  three  sides  respec- 
tively ;  two  expressions  for  each,  one  for  each  of  the  two 
remaining  sides;  that  is  to  be  used  which  contains  the  given 
side. 

From   these   expressions  we   derive   the  values  of  the 
sines  \  thus — 

sin.  A  =  sin.  Ba-.  (197-) 

b 

sin.  A  =  sin.  C  —  .  (19%-) 


sin.  B  =  sin.  A  —.  (I99«) 

•  ci 

sin.  B  =  sin.  C '-.  (200.) 

sin.  C  —  sin.  A  -.  (201.) 

sin.  C  —  sin.  BC-.  (202.) 

477.  —  Oblique  -  Angled   Triangles  :    First   Class.  —  The 

problems  arising  in  the  treatment  of  oblique-angled  trian- 
gles have  been  divided  into  four  classes,  one  of  which,  the 


TO   FIND   THE   TWO   SIDES.  521 

first,  will  here  be  referred  to.  The  problems  of  the  first 
class  are  those  in  which  a  side  and  two  angles  are  given,  to 
find  the  remaining  angle  and  sides. 

As  to  the  required  angle,  since  the  three  angles  of  every 
triangle  amount  to  just  two  right  angles  (Art.  345),  or  180°, 
the  third  angle  may  be  found  simply  by  deducting  the  sum 
of  the  two  given  angles  from  180°. 

For  example :  referring  to  Fig.  318,  if  angle  A  =  18°  and 
angle  B  =  42°,  then  their  sum  is  18  +  42  =  60,  and   180  - 
60  =  120°  =  the  angle  AC  B. 

To  find  the  two  sides :  if  a  be  the  given  side,  then  to  find 
the  side  b  we  have,  equation  (193.)— 

sin.  B 

b  —  a  ~ ; 

sin.  A 

or,  the  side  b  equals  the  product  of  the  side  a  into  the  quo- 
tient obtained  by  a  division  of  the  sine  of  the  angle  opposite 
b  by  the  sine  of  the  angle  opposite  a. 

For  example:  in  a  triangle  (Fig.  318)  in  which  the  angle 
A  =  1 8°,  the  angle  B  —  42°  (and,  consequently  (Art.  345)  the 
angle  C=  120°),  and  the  given  side  a  equals  43  feet;  what 
are  the  lengths  of  the  sides  b  and  cl     Equation  (193.)  gives- 
sin.'  B 

b~a-     --. 
sin.  A 

Performing  the  problem  by  logarithms  (Art.  427),  we 
have — 

Log.   a(=  43)=  I -6334685 
Sin.  B  (-  42°)  =  9-8255109 

£-4589794 
Sin.  A  (=  1 8°)  =  9-4899824 

Log.  b  O  93 « 1 102)  =  i  -9689970. 

Thus  the  side  b  equals  93-1102  feet,  or  93  feet  I  inch  and 
nearly  one  third  of  ah  inch. 


522  TRIGONOMETRY. 

For  the  side  c,  we  have,  equation  (195.)  — 

sin.  C 

c  =  a  —.  —  —  ; 
sin.  A 

or  — 

Log.    *(=  43)  =1-6334685 
Sin.  C(=  120°)  =  9-9375306 

1'  5  709991 
Sin.    A  (—  1  8°)  =  9-4899824 

Log.  c(=  120-508)  —  2-0810167 

or,  the  base  c  equals  120  feet  6  inches  and  one  tenth  of  an 
inch,  nearly.  But  if  instead  of  a  the  side  b  be  given,  then 
for  a  use  equation  (191.),  and  for  c  use  equation  (196.). 

And,  lastly,  if  c  be  the  given  side,  then  for  a  use  equation 
(192.),  and  for  b  use  equation  (194.).  t 

478.  —  Oblique-Angled  Triangles:  Second  Clas§.—  The 
problems  which  comprise  the  second  class  are  those  in  which 
.two  sides  ^ax\&  an  angle  opposite  to  one  of  them  are  given,  to 
find  the  two  remaining  angles  and  the  third  side. 

The  only  requirement  really  needed  here  is  to  find  a 
second  angle  ;  for,  with  this  second  angle  found,  the  problem 
is  reduced  to  one  of  the  fir§t  class  ;  and  the  third  side  may 
then  be  found  under  rules  given  in  Art.  477. 

To  find  a  second  angle,  use  one  of  the  equations  (197.)  to 

(202.). 

For  example  :  in  the  triangle  ABC  (Fig.  318),  let  a  (=  43) 
and  b  (=  93  •  1  1)  be  the  two  given  sides,  and  A,  the  angle  op- 
posite a,  be  the  given  angle  (=  18°).  Then  to  find  the  angle 
B,  we  have  equation  (199.)  —  (selecting  that  which  in  the 
right  hand  member  contains  the  given  angle  and  sides)  — 

sin.  B  =  sin.  A  — 


43 


OBLIQUE-ANGLED   TRIANGLES.  523 

By  logarithms  (Art.  427),  we  have  — 

Log.  sin.  A  (=  1 8°)  =  9-4899824 

"      93-n  =  1.9689970 

1-4589794 

43  =  1-6334685 

"      sin.  5 (=42°)  =  9-8255109 

By  reference  to  the  log.  tables,  the  last  line  of  figures,  as 
above,  is  found  to  be  the  sine  of  42° ;  therefore,  the  required 
angle  B  is  42°.  Then  180°  -  (18°  +  42°)  =  120°  =  the  angle  C. 

With  these  angles,  or  with  any  two  of  them,  the  third 
side  c  may  be  found  by  rules  given  in  Art.  477. 


FIG.  319. 

479. — Oblique-Angled  Triangles  :  Sum  and  Difference 
of  Two  Angles. — Preliminary  to  a  consideration  of  prob- 
lems in  the  third  class  of  triangles,  it  is  requisite  to  show  the 
relation  between  the  sum  and  difference  of  two  angles. 

In  Fig.  319,  let  the  angle  A  JM  and  the  angle  A  JN  be 
the  two  given  angles  ;  and  let  A  J M  be  called  angle  A,  and 
AJN,  angle  B.  Now  the  sum  and  difference  of  the  angles 
may  be  ascertained  by  the  use  of  the  sum  and  difference  of 
the  sines  of  the  angles,  and  by  the  sum  and  difference  of  the 
tangents.  In  the  diagram,  in  which  the  radius  A  J  equals 


524  TRIGONOMETRY. 

unity,  we  have  MP,  the  sine  of  angle  A  (=  A  y  M),  and 
NQ  =  RP,  the  sine  of  angle  B(=  A  y  N).     Then— 

MP-  RP=  MR 

equals  the  difference  of  the  sines  of  the  angles  ;    and  since 
PM'  =  PM— 

PM'+RP  =  RM', 

equals  the  sum  of  the  sines  of  the  angles. 

With  the  radius  JC  describe  the  arc  J  D  E,  and  tangent 
to  this  arc  draw  FH  parallel  with  MM',  or  perpendicular 
to  AB. 

Then  FD  is  the  tangent  of  the  angle  M  C  N,  and  D  H  is 
the  tangent  of  the  angle  NCM'. 

Now  since  an  angle  at  the  circumference  is  equal  to  half 
the  angle  at  the  centre  standing  on  the  same  arc  (Art.  355), 
therefore  the  measure  of  the  angle  M  C  N  is  the  half  of  M  N, 
equals— 


-B). 
Similarly,  we  have  — 


for  the  angle  NCM'. 

Therefore  we  have  for  the  tangent  of  the  angle  M  C  N 


-B\ 

and,  for  the  tangent  of  the  angle  N  C  M'- 
DH  =  tan.  \(A  +  B). 

And,  because  FC  D  and  M  C  R  are  homologous  triangles,  as, 
also,  DCH  and  R'CM',  therefore— 

•   M'  R  :  MR  :  :  DH  :  D  F, 


SUM   AND   DIFFERENCE   OF  TWO   ANGLES.  525 

sin.  A  4-  sin.  B  :  sin.  A  —  sin.  B  :  :  tan.  %(A  +  B)  :  tan.  ±(A  —  B), 

from  which  we  have — 

sin.  A  —  sin.  B  __  tan.  %  (A  —  B)  ,~  . 

sin.  ^4  +  sin.  ./?  .     tan.  %  (A  +  B)' 

To  obtain  a  proper  substitute  for  the  first  member  of  this 
expression  we  have,  equation  (195.)— 

sin.  C 

/•   7>  

C-     —*    LI       ; ~—    . 

sin.  A 
or — 

c  sin.  A  —  a  sin.  C.  (M.) 

We  also  have,  equation  (196.) — 

,sin.  C 

c  =  b  -. , 

sin.  B 

or — 

c  sin.  B  —  b  sin.  C.  (N.) 

These  two  equations,  (M.)  and  (N.),  added,  give — 

c  sin.  A  -t-  c  sin.  B  —  a  sin.  (7  +  b  sin.  C 
or — 

c  (sin.  ^  +  sin.  B)  =  sin.  C(«  +  b).  (P.) 

But,  if  equation  (N.)  be  subtracted  from  equation  (M.),  we 
have — 

c  sin.  A  —  c  sin.  B  —  a  sin.  C  —  b  sin.  C, 
or — 

f(sin.  A  —  sin.  B)  =  A  sin.  C  (a  —  £).'  (R.) 

If  equation  (R.)  be  divided  by  equation  (P.),  we  have— 

<r(sin.  A  —  sin.  B)  __  sin.  C(a  —  b) 
<r(sin.  A  -f  sin.  B)  ~  sin.  C  (a  +  b) ' 


526  TRIGONOMETRY. 

which  reduces  to  — 

sin.  A  —  sin.  B  _  a  —  b 
sin.  A  +  sin.  B  ~  a  +  b' 

The  first  member  of  this  equation  is  identical  with  the  first 
member  of  the  above  equation  (D.),  and  therefore  its  equal, 
the  second  member,  may  be  substituted  for  it  ;  thus— 

a  —  b  _  tan,  j-  (A  -  B) 
a  +~b~  tan.  ^(A~+^)" 

From  which  we  have  —  t 

tan.  $(A—£)  =  tan.  J  (A  +  B)  ~  —  .         (203.) 

We  have  (Art.  431)  the  proposition,  that  if  half  the  differ- 
ence of  two  quantities  be  subtracted  from  half  their  sum,  the 
remainder  will  equal  the  smaller  quantity.  For  example  : 
if  A  represent  the  larger  quantity  and  B  the  smaller,  then  — 


)«>  Jfj  (204.) 

and,  again,  we  also  have  (Art.  431)  — 

B)=A.  (205.) 


480.—  Oblique-Angled   Triangles;  :    Third    <  In**.  —  The 

third  class  of  problems  comprises  all  those  cases  in  which  two 
sides  of  a  triangle  and  their  included  angle  are  given,  to 
find  the  other  side  and  angles. 

In  this  case,  as  in  the  problems  of  the  second  class,  the 
only  requirement  here  is  to  find  a  second  angle  ;  for  then 
the  problem  becomes  one  belonging  to  the  first  class.  But 
the  finding  of  the  second  angle,  in  problems  of  the  third 
class,  is  attended  with  more  computation  than  it  is  in  prob- 
lems of  the  second  class.  The  process  is  as  follows  :  Hav- 
ing one  angle  of  a  triangle,  the  sum  of  the  two  remaining 


OBLIQUE-ANGLED   TRIANGLES.  527 

angles  is  obtained  by  subtracting  the  given  angle  from 
1 80°  —  the  sum  of  the  three  angles. 

Then  with  equation  (203.)  the  difference  of  the  two  angles 
is  obtained.  And  then,  having  the  sum  and  difference  of  the 
two  angles,  either  may  be  found  by  one  of  the  equations 
(204.)  and  (205.). 

For  example :  let  Fig.  320  represent  the  triangle  in 
which  a  (=  36  feet)  and  b(=  27  feet)  are  the  given  sides  ;  and 


C  (=  105°)  the  angle  included  between  the  given  sides,  a  and 
b.     The  sum  of  the  two  angles  A  and  B,  therefore,  will  be  — 

(A  +  £)  =  180-  105  ^75°, 

and  the  half  of  the  sum  of  A  and  B  is  -V  -  37°  30'. 

The  sum  of  the  given  sides  is  36  +  27  =  63,  and  their  dif- 
ference is  36  —  27  =  9. 

Then  from  equation  (203.)  we  have— 


.    tan.  ±(A-B)  =  tan.  37 
Solving  this  by  logs.  (Art.  427),  we  have- 

Log.  tan.  37°  30'  =  9-8849805 
9     =0-9542425 

0-8392230 

63  =  I-79934Q5 

tan.  $(A-  B)(=  6°  15'  20-5")  =  9-0398825 

Thus  half  the  difference  of  A  and  B  is  6°  15'  20-  5",  nearly, 


528  TRIGONOMETRY. 

By  equation  (204.) — 

37°  30' 
6°  15' 20- 5" 

The  difference,  31°  14'  39'$" 
and  by  equation  (205.)— 

37-30 
6.15.20-5 


The  sum,    43.45.20-5  —  A 

From  above,    31.14.39-5  —  ^ 

The  given  angle,  105.  o.  o      —  C 

The  three  angles,  180.  o.  o 

Thus,  by  adding   together  the  three  angles,  the  work  is 
tested  and  proved. 

Having   the   three   angles,  the  third  side  may  now  be 
found  by  the  rule  for  problems  of  the  first  class. 


—  Oblique-  Angled  Triangles:   Fourth  Class.  —  The 

fourth  class  comprises  those  problems  in  which  the  three 
sides  of  the  triangle  are  given,  to  find  the  three  angles. 

The  method  by  which  the  problems  of  the  fourth  class 
are  solved  is  to  divide  the  triangle  into  two  right-angled 
triangles;  then,  by  the  use  of  equation  (129.),  to  find  one 
side  of  one  of  these  triangles,  and  then  with  this  side  to  find 
one  of  the  angles,  then  by  rules  for  the  second  class  prob- 
lems, obtain  the  second  and  third  angles. 

Thus,  from  equation  (129.),  we  have  — 


By  the  relation  of  sines  to  sides  (Art.  476),  we  have  (Fig. 
b  :  g  :  :  sin.  E  :  sin.  F. 


TRIANGLES— FOURTH   CLASS. 


529 


But  the  angle  E  is  a  right  angle,  of  which  the  sine  is  unity, 
therefore^ — 

b  :  g  :  :   I   :  sin.  F  =    -. 


Substituting  for  g  its  value  as  above,  we  have — 


*sin.  F  = 


q-  b) 


2bc 


(206.) 


To  illustrate:  let  a,  b,  c  (Fig.  321)  be  the  three  given  sides 


of  the  triangle  ABC,  respectively  equal  to  12,'  8  and  16  feet. 
With  these,  equation  (206.)  becomes — 

I62-(i2  +  8)(i2-8) 

sin.  F  = *-*»-<         —^ , 

2  x  8  x  16 


sin.  /*  = 


sin.  F 


176 


Solving  this  by  logarithms  (Art.  427),  we  have- 
Log.  176  =  2-2455127 
"     256  =  2-4082400 


Log.  sin.  43°  26'  =  9*8372727 
or,  the  angle  at  F  equals  43°  26',  nearly.    Of  the  triangle 


530 


TRIGONOMETRY. 


A  C E  (Fig.  321),  E  is  a  right  angle,  therefore  the  sum  of  F 
and  A,  the  two  remaining  angles,  equals  90°  (Art.  346). 
Hence,  for  the  angle  at  A,  we  have — 


=90°  -43°  26'  =  46°  34'- 


We  now  have  two  sides  a  and  b  and  A,  an  angle  opposite 
to  one  of  them,  to  find  B,  a  second  angle.  For  this,  equa- 
tion (199.)  is  appropriate.  Thus — 


sin.  B  =  sin.  A  —. 
a 


This  may  be  solved  as  shown  in  Art.  478. 

And,  when  the  second  angle  is  obtained,  the  third  angle 
is  found  by  subtracting  the  sum  of  the  first  and  second  an- 
gles from  1 80°. 

But  to  test  the  accuracy  of  the  work,  it  is  well  to  com- 
pute the  angle  'C  from  the  angle  A,  and  the  sides  a  and  c. 
For  this,  equation  (201.)  will  be  appropriate. 

482.— Trigonometric  Formulae:  Right- Angled  Trian- 
gles.—  For  facility  of  reference  the  formulas  of  previous 


articles  are  here  presented  in  tabular  form.     The  symbols 
referred  to  are  those  of  Fig.  322. 


FORMULA  IN  TABULAR  FORM. 


531 


RIGHT-ANGLED  TRIANGLES. 


GIVEN. 

REQUIRED. 

FORMULA. 

a,    b, 
a,   c, 
b,   c, 

; 

c    =  fV+J*. 

£    =  V(c  +  *)  (c 

-a). 

a   =  V(c  +  ^)  (^ 

-b). 

A, 

A, 

£=  90°  -A. 
A  =  90°  —  ^. 

B,  a, 

*' 

^   =  «  tan.  ^. 

rt: 

"  cos.  B  ' 

B,b, 

*, 

b 

tan.  ^  * 
b 

B.,. 

a, 
b, 

a  =  c  cos.  B, 
b   —  c  sin.  B. 

483. Trigonometrical  Formulae:  First  Class,  Oblique. 

C 


—The  symbols  of  the  formulae  of  the  following-  table  indi- 
cate quantities  represented  in  Fig.  323  by  like  symbols. 


532 


TRIGONOMETRY. 


OBLIQUE-ANGLED  TRIANGLES:    FIRST  CLASS. 


GIVEN,     i  REQUIRED. 

FORMULA. 

; 

A         "D                      /~> 

•fJ.  y     Jj)                          O. 

^=180  —  ^+^. 

A,  C,          B, 

B  =  iSo-A  +  C. 

B,  C,          A, 

A=  180  -  B  +  C. 

A   B  b  \        a 

..sin.  A 

n.t  jjf  t/y  \           u-, 

[ 

A    C  c   \        a 

sin.  B' 
sin.  A 

JJ.)     W',    C-,     |                C*, 

'     .                W-      <s                     ^-,   • 

sm.  (7 

1 
A    B  //           h 

,           sin.  ^ 

A      /7 

f*9   JJj  l*j  \              I/, 

1 

B    C  c            b 

sm.  ^4 
,           sin.  B 

h    —  r 

U)    C-,    t,                   C/j 

sm.  C 

A      C    /r                r 

sin.  £7 

si,  L,,  a,          cy 

\ 

sin.  A 
,sin.  £7 

'•*''.                  ' 

t    —  u  •  .         Ty. 

sm.  j5 

484. — Trigonometrical  Formulae  :  Second  Cla§§,  Oblique. 

—The  symbols  in  the  formulae  of  the  following  table  refer 
to  quantities  represented  in  Fig.  323,  by  like  symbols. 


FORMULAE  FOR  TRIANGLES,   SECOND    CLASS.  533 


OBLIQUE-ANGLED  TRIANGLES:    SECOND  CLASS. 


1 

GIVEN. 

REQUIRED 

FORMULAE. 

B,  a,  b, 

A, 

sin.  A  =  sin.  B  —. 
b 

C,  a,  c, 

A, 

sin.  A  —  sin.  C  —  . 
c 

A,  a,  b, 

By 

sin.  B  =  sin.  A  —. 
a 

C,  b,  c, 

By 

sin.  B  =  sin.  C  —  . 
c 

A,  a,  c, 

c, 

sin.  C  =  sin.  A  —  . 
a 

B,  b,  c, 

Cy 

sin.  C  —  sin.  B-. 

B,  C, 

A, 

A  —  180  —  B  +  C. 

A,  C, 
Ay  By 

By 

B  =  1  80  —  A  +  C. 

C  =  1  80  —  A  +  B. 

For  — 



ay 

See  Formulae,  First  Class. 

534 


TRIGONOMETRY. 


485. — Trigonometrical  Formulae :  Third  Class,  Oblique. 

— The  symbols  in  the  formulae  of  the  following  table  refer 
to  quantities  shown  by  like  symbols  in  Fig.  323. 


OBLIQUE-ANGLED  TRIANGLES:   THIRD  CLASS. 


GIVEN. 

REQUIRED. 

FORMULA. 

C,  a,  b, 

A  +B, 

A-  B, 

A, 

tan.  $(A-B)  =  tan.  $(A+  B)  ~-^. 

B, 

B  =$(A+B)  —  i(A  —  B). 

A,  b,  c, 

C  +B, 
C-B, 

C  +  B  =  1  80  —  A. 

c  —  b 
tan.  4-  (C  —  B)  =  tan.  £  (C  +  B)  —  -—.  . 
c  +  b 

r  ——    1  (  f  JL.    7?'\    _J_  1  /"  f           7?\ 

D           1  (  i^    i      /?\           1  /'/^           J2\ 
-D   ——  "2"  Ivx     T    *'l  —  2  \^   —  **1* 

B,  a,  c, 

C  +  A. 
r      A 

C  +  A  —  180  —  B. 

£    ft 

U                     SI, 

c, 

c  +  a 

A, 

A=t(C+A)  +  i(C-A). 

For  the  remaining  side  consult  formulas  for  the  first 
class. 


486. — Trigonometrical  Formulae:  Fourtli  Class,  Ob- 
lique.— The  symbols  in  the  formulae  of  the  following  table 
refer  to  quantities  shown  by  like  symbols  in  Fig.  321. 


FORMULA   FOR  TRIANGLES,  FOURTH   CLASS.  535 

OBLIQUE-ANGLED  TRIANGLES:    FOURTH  CLASS. 


Given  «,  #,  c,  to  find  A,  B,  C. 


A  =  90  -  F. 
sin.  ^  =  sin.  A  —  . 


^  /i  c 

sin.  c  =  sin.  ^4  -. 


=  180  —  (A  +  B}. 


SECTION    XV.— DRAWING. 

487. — General  Remark§. — A  knowledge  of  the  proper- 
ties and  principles  of  lines  can  best  be  acquired  by  practice. 
Although  the  various  diagrams  throughout  this  work  may 
be  understood  by  inspection,  yet  they  will  be  impressed 
upon  the  mind  with  much  greater  force,  if  they  are  actually 
drawn  out  with  pencil  and  paper  by  the  student.  Science 
is  acquired  by  study — art  by  practice ;  he,  therefore,  who 
would  have  anything  more  than  a  theoretical  (which  must 
of  necessity  be  a  superficial)  knowledge  of  carpentry  and 
geometry,  will  provide  himself  with  the  articles  here  speci- 
fied, and  perform  all  the  operations  described  in  the  fore- 
going and  following  pages.  Many  of  the  problems  may 
appear,  at  the  first  reading,  somewhat  confused  and  intricate  ; 
but  by  making  one  line  at  a  time,  according  to  the  explana- 
tions, the  student  will  not  only  succeed  in  copying  the  fig- 
ures correctly,  but  by  ordinary  attention  will  learn  the 
principles  upon  which  they  are  based,  and  thus  be  able  to 
make  them  available  in  any  unexpected  case  to  which  they 
may  apply. 

488.— Articles  Required.  — The  following  articles  are 
necessary  for  drawing,  viz. :  a  drawing-board,  paper,  draw- 
ing-pins or  mouth-glue,  a  sponge,  a  T-square,  a  set-square, 
two  straight-edges,  or  flat  rulers,  a  lead  pencil,  a  piece  of 
india-rubber,  a  cake  of  india-ink,  a  set  of  drawing-instru- 
ments, and  a  scale  of  equal  parts. 

489. — The  Drawing-Board. — The  size  of  the  drawing- 
board  must  be  regulated  according  to  the  size  of  the  draw- 
ings which  are  to  be  made  upon  it.  Yet  for  ordinary  prac- 
tice, in  learning  to  draw,  a  board  about  fifteen  by  twenty 
inches,  and  one  inch  thick,  will  be  found  large  enough,  and 


DRAWING   PAPER.  537 

more  convenient  than  a  larger  one.  This  board  should  be 
well  seasoned,  perfectly  square  at  the  corners,  and  without 
clamps  on  the  ends.  A  board  is  better  without  clamps, 
because  the  little  service  they  are  supposed  to  render  by 
preventing  the  board  from  warping  is  overbalanced  by  the 
consideration  that  the  shrinking  of  the  panel  leaves  the 
ends  of  the  clamps  projecting  beyond  the  edge  of  the  board, 
and  thus  interfering  with  the  proper  working  of  the  stock 
of  the  T-square.  When  the  stuff  is  well-seasoned,  the  warp- 
ing of  the  board  will  be  but  trifling ;  and  by  exposing  the 
rounding  side  to  the  fire,  or  to  the  sun,  it  may  be  brought 
back  to  its  proper  shape. 

490. — Drawing-Paper. —  For  mere  line  drawings,  it  is 
unnecessary  to  use  the  best  drawing-paper  ;  and  since,  where 
much  is  used,  the  expense  will  be  considerable,  it  is  desirable 
for  economy  to  procure  a  paper  of  as  low  a  price  as  will  be 
suitable  for  the  purpose.  The  best  paper  is  made  in  Eng- 
land and  water-marked  "  Whatman."  This  is  a  hand-made 
paper.  There  is  also  a  machine-made  paper  at  about  half- 
price,  and  the  manilla  paper,  of  various  tints  of  russet  color, 
is  still  less  in  price.  These  papers  are  of  the  various  sAzes 
needed,  and  are  quite  sufficient  for  ordinary  drawings. 

49 1. — To  Secure  the  Paper  to  the  Board. — A  drawing- 
pin  is  a  small  brass  button,  having  a  steel  pin  projecting  from 
the  underside.  By  having  one  of  these  at  each  corner,  the 
paper  can  be  fixed  to  the  board  ;  but  this  can  be  done  in  a 
better  manner  with  moutJi-glue.  The  pins  will  prevent  the 
paper  from  changing  its  position  on  the  board ;  but,  more 
than  this,,  the  glue  keeps  the  paper  perfectly  tight  and 
smooth,  thus  making  it  so  much  the  more  pleasant  to  work 
on. 

To  attach  the  paper  with  mouth-glue,  lay  it  with  the 
bottom  side  up,  on  the  board  ;  and  with  a  straight-edge  and 
penknife  cut  off  the  rough  and  uneven  edge.  With  a 
sponge  moderately  wet  rub  all  the  surface*  of  the  paper, 
except  a  strip  around  the  edge  about  half  an  inch  wide.  As 
soon  as  the  glistening  of  the  water  disappears  turn  the  sheet 


DRAWING.       • 

over  and  place  it  upon  the  board  just  where  you  wish  it 
glued.  Commence  upon  one  of  the  longest  sides,  and  pro- 
ceed thus :  lay  a  flat  ruler  upon  the  paper,  parallel  to  the 
edge,  and  within  a  quarter  of  an  inch  of  it.  With  a  knife, 
or  anything  similar,  turn  up.  the  edge  of  the  paper  against 
the  edge  of  the  ruler,  and  put  one  end  of  the  cake  of  mouth- 
glue  between  your  lips  to  dampen  it.  Then  holding  it  up- 
right, rub  it  against  and  along  the  entire  edge  of  the  paper 
that  is  turned  up  against  the  ruler,  bearing  moderately 
against  the  edge  of  the  ruler,  which  must  be  held  firmly 
with  the  left  hand.  Moisten  the  glue  as  often  as  it  becomes 
dry,  until  a  sufficiency  of  it  is  rubbed  on  the  edge  of  the 
paper.  -Take  away  the  ruler,  restore  the  turned-up  edge  to 
the  level  of  the  board,  and  lay  upon  it  a  strip  of  pretty  stiff 
paper.  By  rubbing  upon  this,  not  very  hard  but  pretty 
rapidly,  with  the  thumb-nail  of  the  right  hand,  so  as  to  cause 
a  gentle  friction  and  heat  to  be  imparted  to  the  glue  that  is 
on  the  edge  of  the  paper,  you  will  make  it  adhere  to  the 
board.  The  other  edges  in  succession  must  be  treated  in 
the  same  manner. 

Some  short  distances  along  one  or  more  of  the  edges 
may  afterward  be  found  loose  ;  if  so,  the  glue  must  again 
be  applied,  and  the  paper  rubbed  until  it  adheres.  The 
board  must  then  be  laid  away  in  a  warm  or  dry  place ;  and 
in  a  short,  time  the  surface  of  the  paper  will  be  drawn  out, 
perfectly  tight  and  smooth,  and  ready  for  use.  The  paper 
dries  best  when  the  board  is  laid  level.  When  the  drawing 
is  finished  lay  a  straight-edge  upon  the  paper  and  cut  it 
from  the  board,  leaving  the  glued  strip  still  attached.  This 
may  afterward  be  taken  off  by  wetting  it  freely  with  the 
sponge,  which  will  soak  the  glue  and  loosen  the  paper.  Do 
this  as  soon  as  the  drawing  is  taken  off,  in  order  that  the 
board  may  be  dry  when  it  is  wanted  for  use  again.  Care 
must  be  taken  that,  in  applying  the  glue,  the  edge  of  the 
paper  does  not  become  damper  than  the  rest ;  if  it  should, 
the  paper  must  be  laid  aside  to  dry  (to  use  at  another  time) 
and  another  sheet  be  used  in  its  place. 

Sometimes,  especially  when  the  draAving-board  is  new, 
the  paper  will  not  stick  very  readily  ;  but  by  persevering 


THE   T-SQUARE.  539 

this  difficulty  may  be  overcome.  In  the  place  of  the  mouth- 
glue  a  strong  solution  of  gum-arabic  may  be  used,  and  on 
some  accounts  is  to  be  preferred  ;  for  the  edges  of  the  paper 
need  not  be  kept  dry,  and  it  adheres  more  readily.  Dissolve 
the  gum  in  a  sufficiency  of  warm  water  to  make  it  of  the 
consistency  of  linseed-oil.  It  must  be  applied  to  the  paper 
with  a  brush,  when  the  edge  is  turned  up  against  the  ruler, 
as  was  described  for  the  mouth-glue.  If  two  drawing-boards 
are  used,  one  may  be  in  use  while  the  other  is  laid  away  to 
dry;  and  as' they  may  be  cheaply  made,  it  is  advisable  to 
have  two.  The  drawing-board  having  a  frame  around  it, 
commonly  called  a  panel  board,  may  afford  rather  more 
facility  in  attaching  the  paper  when  this  is  of  the  size  to 


FIG.  324. 

suit ;  yet  it  has  objections  which  overbalance  that  consid- 
eration. 

492. — The  T-Square. — A  T-square  of  mahogany,  at  once 
simple  in  its  construction  and  affording  all  necessary  service, 
may  be  thus  made :  let  the  stock  or  handle  be  seven  inches 
long,  two  and  a  quarter  inches  wide,  and  three  eighths  of  an 
inch  thick ;  the  blade,  twenty  inches  long  (exclusive  of  the 
stock),  two  inches  wide,  and  one  eighth  of  an  inch  thick.  In 
joining  the  blade  to  the  stock,  a  very  firm  and  simple  joint 
may  be  made  by  dovetailing  it — as  shown  at  Fig.  324. 

493. — Tlie  Set-Square. — The  set-square  is  in  the  form  of 
a  right-angled  triangle  ;  and  is  commonly  made  of  mahogany, 


540  DRAWING. 

one  eighth  of  an  inch  in  thickness.  The  size  that  is  most 
convenient  for  general  use  is  six  inches  and  three  inches 
respectively  for  the  sides  which  contain  the  right  angle, 
although  a  particular  length  for  the  sides  is  by  no  means 
necessary.  Care  should  be  taken  to  have  the  square  corner 
exactly  true.  This,  as  also  the  T-square  and  rulers,  should 
have  a  hole  bored  through  them,  by  which  to  hang  them 
upon  a  nail  when  not  in  use. 

494. — The  Rulers.  —  One  of  the  rulers  may  be  about 
twenty  inches  long,  and  the  other  six  inches.  The  pencil 
ought  to  be  hard  enough  to  retain  a  fine  point,  and  yet  not 
so  hard  as  to  leave  ineffaceable  marks.  It  should  be  used 
lightly,  so  that  the  extra  marks  that  are  not  needed  when 
the  drawing  is  inked,  may  be  easily  rubbed  off  with  the 
rubber.  The  best  kind  of  india-ink  is  that  which  will  easily 
rub  off  upon  the  plate  ;  and,  when  the  cake  is  rubbed  against 
the  teeth,  will  be  free  from  grit. 

495. — The  Instruments. — The  drawing-instruments  may 
be  purchased  of  mathematical  instrument  makers  at  various 
prices ;  from  one  to  one  hundred  dollars  a  set.  In  choosing 
a  set,  remember  that  the  lowest  price  articles  are  not  always 
the  cheapest.  A  set,  comprising  a.  sufficient  number  of 
instruments  for  ordinary  use,  well  made  and  fitted  in  a  ma- 
hogany box,  may  be  purchased  of  the  mathematical  instru- 
ment makers  in  New  York  for  four  or  five  dollars.  But  for 
permanent  use  those  which  come  at  ten  or  twelve  dollars 
will  be  found  to  be  better. 

496. — The  Scale  of  Equal  Parts. — The  best  scale  of 
equal  parts  for  carpenters'  use,  is  one  that  has  one  eighth, 
three  sixteenths,  one  fourth,  three  eighths,  one  half,  five 
eighths,  three  fourths,  and  seven  eighths  of  an  inch,  arid  one 
inch,  severally  divided  into  twelfths,  instead  of  being  divided, 
as  they  usually  are,  into  tenths.  By  this,  if  it  be  required 
to  proportion  a  drawing  so  that  every  foot  of  the  object 
represented  will  upon  the  paper  measure  one  fourth  of  an 
inch*,  use  that  part  of  the  scale  which  is  divided  into  one 


THE   SET-SQUARE.  541 

fourths  of  an  inch,  taking  for  every  foot  one  of  those  divis- 
ions, and  for  every  inch  one  of  the  subdivisions  into  twelfths ; 
and  proceed  in  like  manner  in  proportioning  a  drawing  to 
any  of  the  other  divisions  of  the  scale.  An  instrument  in 
the  form  of  a  semi-circle,  called  a  protractor,  and  used  for 
laying  down  and  measuring  angles,  is  of  much  service  to 
surveyors,  and  occasionally  to  carpenters. 

497. — The  U§e  of  the  Set-Square. — In  drawing  parallel 
lines,  when  they  are  to  be  parallel  to  either  side  of  the 
board,  use  the  T-square ;  but  when  it  is  required  to  draw 
lines  parallel  to  a  line  which  is  drawn  in  a  direction  oblique 


FIG.  325. 

to  either  side  of  the  board,  the  set-square  must  be  used. 
Let  ab  (Fig.  325)  be  a  line,  parallel  to  which  it  is  desired  to 
draw  one  or  more  lines.  Place  any  edge,  as  c d,  of  the  set- 
square  even  with  said  line  ;  then  place  the  ruler  gh  against 
one  of  the  other  sides,  as  ce,  and  hold  it  firmly  ;  slide  the 
set-square  along  the  edge  of  the  ruler  as  far  as  it  is  desired, 
as  at  /;  and  a  line  drawn  by  the  edge  *'/  will  be  parallel 
to  a  b. 

To  draw  a  line,  as  kl  (Fig.  326),  perpendicular  to  another, 
as  a  b,  set  the  shortest  edge  of  the  set-square  at  the  line  a  b ; 
place  the  ruler  against  the  longest  side  (the  hypothenuse  of 
the  right-angled  triangle);  hold  the  ruler  firmly,  and  slide 
the  set-square  along  until  the  side  ed  touches  the  point  k\ 
then  the  line  Ik,  drawn  by  it,  will  be  perpendicular  to  ab. 


542  DRAWING. 

In  like  manner,  the  drawing  of  other  problems  may  be  facil- 
itated, as  will  be  discovered  in  using  the  instruments. 

498. — Directions  for  Drawing. — In  drawing  a  problem, 
proceed,  with  the  pencil  sharpened  to  a  point,  to  lay  down 
the  several  lines  until  the  whole  figure  is  completed,  ob- 
serving to  let  the  lines  cross  each  other  at  the  several  angles, 
instead  of  merely  meeting.  By  this,  the  length  of  every 
line  will  be  clearly  defined.  With  a  drop  or  two  of  water, 
rub  one  end  of  the  cake  of  ink  upon  a  plate  or  saucer,  until 
a  sufficiency  adheres  to  it.  Be  careful  to  dry  the  cake  of 


FIG  326. 

ink  ;  because  if  it  is  left  wet  it  will  crack  and  crumble  in 
pieces.  With  an  inferior  camel's-hair  pencil  add  a  little 
water  to  the  ink  that  was  rubbed  on  the  plate,  and  mix  it 
well.  It  should  be  diluted  sufficiently  to  flow  freely  from 
the  pen,  and  yet  be  thick  enough  to  make  a  black  line.  With 
the  hair  pencil  place  a  little  of  the  ink  between  the  nibs  of 
the  drawing-pen,  and  screw  the  nibs  together  until  the  pen 
makes  a  fine  line.  Beginning  with  the  curved  lines,  proceed 
to  ink  all  the  lines  of  the  figure,  being  careful  now  to  make 
every  line  of  its  requisite  length.  If  tl.ey  are  a  trifle  too 
short  or  too  long  the  drawing  will  have  a  ragged  appear- 
ance ;  and  this  is  opposed  to  that  neatness  and  accuracy 
which  is  indispensable -to  a  good  drawing.  When  the  ink 
is  dry  efface  the  pencil-marks  with  the  india-rubber.  If  the 


PUTTING  THE   DRAWING   IN   INK.  543 

pencil  is  used  lightly  they  will  all  rub  off,  leaving  those  lines 
only  that  were  inked. 

In  problems  all  auxiliary  lines  are  drawn  light ;  while  the 
lines  given  and  those  sought,  in  order  to  be  distinguished  at 
a  glance,  are  made  much  heavier.  The  heavy  lines  are 
made  so  by  passing  over  them  a  second  time,  having  the 
nibs  of  the  pen  separated  far  enough  to  make  the  lines  as 
heavy  as  desired.  If  the  heavy  lines  are  made  before  the 
drawing  is  cleaned  with  the  rubber  they  will  not  appear  so 
black  and  neat,  because  the  india-rubber  takes  away  part 
of  the  ink.  If  the  drawing  is  a  ground-plan  or  elevation  of 
a  house,  the  shade-lines,  as  they  are  termed,  should  not  be 
put  in  until  the  drawing  is  shaded  ;  as  there  is  danger  of  the 
heavy  lines  spreading  when  the  brush,  in  shading  or  color- 
ing, passes  over  them.  If  the  lines  are  inked  with  common 
writing-ink  they  will,  however  fine  they  may  be  made,  be 
subject  to  the  same  evil ;  for  which  reason  india-ink  is  the 
only  kind  to  be  used. 


SECTION  XVI.— PRACTICAL    GEOMETRY. 

499. — Definitions. — Geometry  treats  of  the  properties  of 
magnitudes. 

A  point  has  neither  length,  breadth,  nor  thickness. 

A  line  has  length  only. 

Superficies  has  length  and  breadth  only. 

A  plane  is  a  surface,  perfectly  straight  and  even  in  every 
direction ;  as  the  face  of  a  panel  when  not  warped  nor 
winding. 

A  solid  has  length,  breadth,  and  thickness. 

A  right,  or  straight,  line  is  the  shortest  that  can  be  drawn 
between  two  points. 

Parallel  lines  are  equidistant  throughout  their  length. 


FIG.  327.  FIG.  328.  FIG.  329. 

An  angle  is  the  inclination  of  two  lines  towards  one  an- 
other (Fig.  327). 

A  right  angle  has  one  line  perpendicular  to  the  other 
(Fig.  328). 

An  oblique  angle  is  either  greater  or  less  than  a  right 
angle  (Figs.  327  and  329). 

An  acute  angle  is  less  than  a  right  angle  (Fig.  327). 

An  obtuse  angle  is  greater  than  a  right  angle  (Fig.  329). 

When  an  angle  is  denoted  by  three  letters,  the  middle 
one,  in  the  order  they  stand,  denotes  the  angular  point,  and 
the  other  two  the  sides  containing  the  angle  ;  thus,  let  a,  b,  c 
(Fig.  327)  be  the  angle,  then  b  will  be  the  angular  point,  and 
ab  and  be  will  be  the  two  sides  containing  that  angle. 


TRIANGLES   AND   RECTANGLES. 


545 


A  triangle -is  a  superficies  having  three  sides  and  angles 
(Figs.  330,  331,  332,  and  333). 

An  equilateral  triangle  has  its  three  sides  equal  (Fig.  330). 
An  isosceles  triangle  has  only  two  sides  equal  (Fig.  331). 


FIG.  330. 


FIG  331. 


A  scalene  triangle  has  all  its  sides  unequal  (Fig.  332). 
A  right-angled  triangle  has  one  right  angle  (Fig.  333). 
An  acute-angled  triangle  has  all  its  angles  acute  (Figs.  330 
and  331). 


FIG.  332. 


FIG.  333. 


An  obtuse-angled  triangle  has  one  obtuse  angle  (Fig.  332). 
A  quadrangle  has  four  sides  and  four  angles  (Figs.  334  to 

339). 

A  parallelogram  is  a  quadrangle  having  its  opposite  sides 

parallel  (Figs.  334  to  337). 


FIG.  334- 


FIG.  335. 


A  rectangle  is  a  parallelogram,  its  angles  being  right 
angles  (Figs.  334  and  335). 

A  square  is  a  rectangle  having  equal  sides  (Fig.  334). 

A  rhombus  is  an  equilateral  parallelogram  having  oblique 
angles  (Fig.  336). 


546  PRACTICAL   GEOMETRY. 

A  rhomboid  is   a   parallelogram    having  oblique  angles 

(Fig-  337). 

A  trapezoid  is  a  quadrangle  having  only  two  of  its  sides 
parallel  (Fig.  338). 


FIG.  336.  FIG.  337. 

A  trapezium  is  a  quadrangle  which  has  no  two  of  its  sides 
parallel  (Fig.  339). 

A  polygon  is  a  figure  bounded  by  right  lines. 

A  regular  polygon  has  its  sides  and  angles  equal. 

An  irregular  polygon  has  its  sides  and  angles  unequal. 


FIG.  338.  FIG.  339. 

A  trigon  is  a  polygon  of  three  sides  (Figs.  330  to  333) ;  a 
tetragon  has  four  sides  (Figs.  334  to  339) ;  a  pentagon  has  five 
(Fig.  340) ;  a  hexagon  six  (Fig.  341) ;  a  heptagon  seven  (/%-. 
342) ;  an  octagon  eight  (/r/£~.  343) ;  a  nonagon  nine  ;  a  decagon 
ten  ;  an  undecagon  eleven  ;  and  a  dodecagon  twelve  sides. 


FIG.  340.  FIG.  341.  FIG.  342.  FIG.  343. 

A  circle  is  a  figure  bounded  by  a  curved  line,  called  the 
circumference,  which  is  everywhere  equidistant  from  a  cer- 
tain point  within,  called  its  centre. 

The  circumference  is  also  called  the  periphery,  and  some- 
times the  circle. 


PARTS   OF   THE   CIRCLE. 


547 


The  radius  of  a  circle  is  a  right  line  drawn  from  the 
centre  to  any  point  in  the  circumference  (ab,  Fig.  334). 

All  the  radii  of  a  circle  are  equal. 

The  diameter  is  a  right  line  passing  through  the  centre, 
and  terminating  at  two  opposite  points  in  the  circumference. 
Hence  it  is  twice  the  length  of  the  radius  (cd.  Fig.  344.) 


FIG.  344. 


An  arc  of  a  circle  is  a  part  of  the  circumference  (cb,  or 
bed,  Fig.  344). 

A  chord  is  a  right  line  joining  the  extremities  of  an  arc 
(b  d,  Fig.  344). 

A  segment  is  any  part  of  a  circle  bounded  by  an  arc  and 
its  chord  (A,  Fig.  344). 


FIG.  345. 


A  sector  is  any  part  of  a  circle  bounded  by  an  arc  and 
two  radii,  drawn  to  its  extremities  (B,  Fig.  344). 

A  quadrant,  or  quarter  of  a  circle,  is  a  sector  having  a 
quarter  of  the  circumference  for  its  arc  (C,  Fig.  344). 

A  tangent  is  a  right  line  which,  in  passing  a  curve, 
touches,  withont  cutting  it  (fg,  Fig.  344). 


PRACTICAL   GEOMETRY. 


A  cone  is  a  solid  figure  standing  upon  a  circular  base  di- 
minishing in  straight  lines  to  a  point  at  the  top,  called  its 
vertex  (Fig:  345). 

The  axis  of  a  cone  is  a  right  line  passing  through  It, 
from  the  vertex  to  the  centre  of  the  circle  at  the  base. 

An  ellipsis  is  described  if  a  cone  be  cut  by  a  plane,  not 
parallel  to  its  base,  passing  quite  through  the  curved  surface 
(a  b,  Fig.  346). 

A  parabola  is  described  if  a  cone  be  cut  by  a  plane,  par- 
allel to  a  plane  touching  the  curved  surface  (c  d,  Fig.  346 — 
cd  being  parallel  to  f  g\ 

An  hyperbola  is  described  if  a  cone  be  cut  by  a  plane, 


FIG.  347.     t 

parallel  to  any  plane  within  the  cone  that  passes  through  its 
vertex  (e/t,  Fig.  346). 

Foci  are  the  points  at  which  the  pins  are  placed  in  de- 
scribing an  ellipse  (see  Art.  548,  and  /,/,  Fig.  347). 

The  transverse  axis  is  the  longest  diameter  of  the  ellipsis 
(a  b,  Fig.  347). 

The  conjugate  axis  is  the  shortest  diameter  of  the  ellipsis  ; 
and  is,  therefore,  at  right  angles  to  the  transverse  axis  (cd, 

Fig.  347). 

The  parameter  is  a  right  line  passing  through  the  focus 
of  an  ellipsis,  at  right  angles  to  the  transverse  axis,  and  ter- 
minated by  the  curve  (gk  and  gt,  Fig.  347). 


RIGHT   LINES   AND  ANGLES. 


549 


A  diameter  of  an  ellipsis  is  any  right  line  passing  through 
the  centre,  and  terminated  by  the  curve  (kl,  or  ;;/ n,  Fig.  347). 

A  diameter  is  conjugate  to  another  when  it  is  parallel  to  a 
tangent  drawn  at  the  extremity  of  that  other — thus,  the  di- 
ameter mn  (Fig.  347)  being  parallel  to  the  tangent  op,  is 
therefore  conjugate  to  the  diameter  kl. 

A  double  ordinate  is  any  right  line,  crossing  a  diameter  of 
an  ellipsis,  and  drawn  parallel  to  a  tangent  at  the  extremity 
of  that  diameter  (it,  Fig.  347). 

A  cylinder  is  a  solid  generated  by  the%  revolution  of  a 
right-angled  parallelogram,  or  rectangle,  about  one  of  its 


FIG.  348. 


FIG.  349. 


sides ;  and  consequently  the  ends  of  the  cylinder  are  equal 
circles  (Fig.  348). 

The  axis  of  a  cylinder  is  a  right  line  passing  through  it 
from  the  centres  of  the  two  circles  which  form  the  ends. 

A  segment  of  a  cylinder  is  comprehended  under  three 
planes,  and  the  curved  surface  of  the  cylinder.  Two  of 
these  are  segments  of  circles ;  the  other  plane  is  a  parallelo- 
gram, called  by  way  of  distinction,  the  plane  of  tJie  segment. 
The  circular  segments  are  called  the  ends  of  the  cylinder 

(Fig.  349)- 

PROBLEMS. 


RIGHT   LINES   AND   ANGLES. 


500. — To  Bi§ect  a  Line. — Upon  the  ends  of  the  line  ab 
(Fig.  350)  as  centres,  with  any  distance  for  radius  greater 
than  half  ab,  describe  arcs  cutting  each  other  in 


550 


PRACTICAL  GEOMETRY. 


draw  the  line  cd,  and  the  point  e,  where  it  cuts  ab}  will  be 
the  middle  of  the  line  ab. 

In  practice,  a  line  is  generally  divided  with  the  com- 
passes, or  dividers  ;  but  this  problem  is  useful  where  it  is 


desired  to  draw,  at  the  middle  of  another  line,  one  at  right 
angles  to  it.     (See  Art.  514.) 

501. — To  Erect  a  Perpendicular. — From  the  point  a 
(Fig.  351)  set  off  any  distance,  as  ab,  and  the  same  distance 
from  a  to  c ;  upon  c,  as  a  centre,  with  any  distance  for  radius 
greater  than  ca,  describe  an  arc  at  d\  upon  b,  with  the  same 


FIG.  351. 

radius,  describe  another  at  d\  join  d  and  a,  and  the  line  da 
will  be  the  perpendicular  required. 

This,  and  the  three  following  problems,  are  more  easily 
performed  by  the  use  of  the  set-square  (see  Art.  493).  Yet 
they  are  useful  when  the  operation  is  so  large  that  a  set- 
square  cannot  be  used. 


TO    ERECT  A  PERPENDICULAR. 


551 


502. — To  let  Fall  a  Perpendicular. — Let  a  (Fig.  352)  be 
the  point  above  the  line  be  from  which  the  perpendicular  is 
required  to  fall.  Upon  a,  with  any  radius  greater  than  ad, 
describe  an  arc,  cutting  be  at  ^and  /;  upon  the  points  e  and 
/,  with  any  radius  greater  than  ed,  describe  arcs,  cutting 


FIG.  352. 

each  other  at  g\    join  a  and  g,  and  the  line  ad  will  be  the 
perpendicular  required. 

503. — To  Erect  a  Perpendicular  at  the  End  of  a  Line. 

—Let  a  (Fig.  353),  at  the  end  of  the  line  c  a,  be  the  point  at 
which  the  perpendicular  is  to  be  erected.  Take  any  point, 
as  by  above  the  line  ca,  and  with  the  radius  ba  describe  the 
arc  dae;  through  d  and  b  draw  the  line  de\  join  e  and  a, 
then  e  a  will  be  the  perpendicular  required. 


FIG.  353- 

The  principle  here  made  use  of  is  a  very  important  one, 
and  is  applied  in  many  other  cases  (see  Art.  510,  3d,  and  Art. 
513.  For  proof  of  its  correctness,  see  Art.  352). 

A  second  method.  Let  b  (Fig.  354),  at  the  end  of  the  line 
a  b,  be  the  point  at  which  it  is  required  to  erect  a  perpendic- 
ular. Upon  b,  with  any  radius  less  than  b  a,  describe  the  arc 
ced\  upon  ct  with  the  same  radius,  describe  the  small  arc  at*/ 


552 


PRACTICAL   GEOMETRY. 


and  upon  c,  another  at  d ;  upon  e  and  d,  with  the  same  or  any 
other  radius  greater  than  half  e  d,  describe  arcs  intersecting 
at  f'  join  /and  b,  and  the  line  fb  will  be  the  perpendicular 
required.  This  method  of  erecting  a  perpendicular,  and 
that  of  the  following  article,  depend  for  accuracy  upon  the 


c  b 

FIG.  354. 


fact  that  the  side  of  a  hexagon  is  equal  to  the  radius  of  the 
circumscribing  circle. 

A  third  method.  Let  b  (Fig.  355)  be  the  given  point  at 
which  it  is  required  to  erect  a  perpendicular.  Upon  b,  with 
any  radius  less  than  ba,  describe  the  quadrant  def\  upon  d, 
with  the  same  radius,  describe  an  arc  at  e,  and  upon  e  an- 
other at  c ;  through  d  and  e  draw  dc,  cutting  the  arc  in  c ; 
join  c  and  3,  then  cb  will  be  the  perpendicular  required. 


d  b 

FIG.  355. 


This  problem  can  be  solved  by  the  six,  eight  and  ten  rule, 
as  it  is  called,  which  is  founded  upon  the  same  principle  as 
the  problems  at  Arts.  536,  537,  and  is  applied  as  follows: 
let  ad  (Fig.  353)  equal  eight,  and  ae,  six ;  then,  if  de  equals 
ten,  the  angle  cad  is  a  right  angle.  Because  the  square  of 
six  and  that  of  eight,  added  together,  equal  the  square  of 


EQUAL  ANGLES.  553 

ten,  thus  :  6  x  6  =  36,  and  S  x  8  —  64;  36  +  64  =  100,  and 
10  x  10  =  100.  Any  sizes,  taken  in  the  same  proportion,  as 
six,  eight  and  ten,  will  produce  the  same  effect ;  as  3,  4  and 
5,  or  12,  1 6  and  20.  (See  Art.  536.) 

By  the  process  shown  at  Fig.  353,  the  end  of  a  board  may 
be  squared  without  a  carpenters'-square.  All  that  is  neces- 
sary is  a  pair  of  compasses  and  a  ruler.  Let  ca  be  the  edge 
of  the  board,  and  a  the  point  at  which  it  is  required  to  be 
squared.  Take  the  point  b  as  near  as  possible  at  an  angle 
of  forty-five  degrees,  or  on  a  mitre-line  from  a,  and  at  about 
the  middle  of  the  board.  This  is  not  necessary  to  the  work- 
ing of  the  problem,  nor  does  it  affect  its  accuracy,  but  the 
result  is  more  easily  obtained.  Stretch  the  compasses  from 
b  to  a,  and  then  bring  the  leg  at  a  around  to  d\  draw  a  line 
from  d,  through  b,  out  indefinitely ;  take  the  distance  db  and 
place  it  from  b  to  c ;  join  e  and  a ;  then  ca  will  be  at  right 
angles  to  c  a.  In  squaring  the  foundation  of  a  building,  or 
laying  out  a  garden,  a  rod  and  chalk-line  may  be  used  in- 
stead of  compasses  and  ruler. 

504. — To  let  Fall  a  Perpendicular  near  the  End  of  a 
Line. — Let  e  (Fig.  353)  be  the  point  above  the  line  c  a,  from 
which  the  perpendicular  is  required  to  fall.  From  e  draw 
any  line,  as  e  d,  obliquely  to  the  line  ca;  bisect  edai  b\  upon 
b,  with  the  radius  be,  describe  the  arc  ead\  join  e  and  #; 
then  ea  will  be  the  perpendicular  required. 

505. — To  Make  an  Angle  (a§  edf,  Fig.  356*)  Equal  to  a 
Given  Angle  (as  b  a  c).— From  the  angular  point  a,  with  any 


FIG.  356. 

radius,  describe  the  arc  b  c ;  and  with  the  same  radius,  on 
the  line  dc,  and  from  the  point  d,  describe  the  arc/^-;  take 
the  distance  be,  and  upon  gt  describe  the  small  arc  at/; 


554 


PRACTICAL   GEOMETRY. 


join  f  and  d\  and  the  angle  edf  will  be  equal  to  the  angle 
bac. 

If  the  given  line  upon  which  the  angle  is  to  be  made  is 
situated  parallel  to  the.  similar  line  of  the  given  angle,  this 
may  be  performed  more  readily  with  the  set-square.  (See 
Art.  497.) 

506. — To  Bisect  an  Angle. — Let  a  be  (Fig.  357)  be  the 
angle  to  be  bisected.  Upon  b,  with  any  radius,  describe  the 


FIG.  357- 

arc  a  c ;  upon  a  and  c,  with  a  radius  greater  than  half  a  c, 
describe  arcs  cutting  each  other  at  d;  join  b  and  d\  and  bd 
will  bisect  the  angle  a  be,  as  was  required. 

This  problem  is  frequently  made  use  of  in  solving  other 
problems ;  it  should  therefore  be  well  impressed  upon  the 
memory. 

507 — To  Trisect  a  Right  Angle.— Upon  a  (Fig.  358), 
with  any  radius,  describe  the  arc  b  c ;  upon  b  and  c,  with  the 


FIG.  358. 


same  radius,  describe  arcs  cutting  the  arc  be  at  d  and  e\ 
from  d  and  e  draw  lines  to  a,  and  they  will  trisect  the  angle, 
as  was  required. 


TO    DIVIDE   A   GIVEN   LINE 

The  truth  of  this  is  made  evident  by  the  following  oper- 
ation :  divide  a  circle  into  quadrants ;  also,  take  the  radius 
in  the  dividers,  and  space  off  the  circumference.  This  will 
divide  the  circumference  into  just  six  parts.  A  semi-circum- 
ference, therefore,  is  equal  to  three,  and  a  quadrant  to  one 
and  a  half  of  those  parts.  The  radius,  therefore,  is  equal  to 
two  thirds  of  a  quadrant ;  and  this  is  equal  to  a  right  angle. 

508. — Through  a  Given  Point,  to  Draw  a  Line  Parallel 
to  a  Given  Line. — Let  a  (Fig.  359)  be  the  given  point,  and 


FIG.  359. 

be  the  given  line.     Upon  any  point,  as  d,  in  the  line  be,  with 
the  radius  da,  describe  the  arc  ac\  upon  a,  with  the  same 
radius,  describe  the  arc  de\  make  de  equal  to  ac\  through 
e  and  a  draw  the  line  ea,  which  will  be  the  line  required. 
This  is  upon  the  same  principle  as  Art.  505. 

509.  —  To  Divide  a  Given  Line  into  any  Number  of 
Equal  Part§. — Let  a  b  (Fig.  360)  be  the  given  line,  and  5  the 
number  of  parts.  Draw  ac  at  any  angle  to  a  b ;  on  ac,  from 


J  A' 

FIG.  360. 


a,  set  off  five  equal  parts  of  any  length,  as  at  i,  2,  3,  4  and  c ; 
join  c  and  b\  through  the  points  I,  2,  3,  and  4,  draw  I  e ,  2/, 
3^ and  4//,  parallel  to  cb\  which  will  divide  the  line  ab,  as 
was  required. 


556 


PRACTICAL   GEOMETRY. 


The  lines  ab  and  ac  are  divided  in  the  same  proportion. 
(See  Art.  542.) 


THE   CIRCLE. 


510. — To  Find  the  Centre  of  a  Circle. — Draw  any  chord, 
as  ab  (Fig.  361),  and  bisect  it  with  the  perpendicular  cd\  bi- 


sect cd  with  the  line  e  f,  as  at  g\  then  g  is  the  centre,  as  was 
required. 

A  second  method.     Upon  any  two  points  in  the  circumfer- 
ence nearly  opposite,  as  a  and  b  (Fig.  362),  describe  arcs  cut- 


ting each  other  at  c  and  d;  take  aay  other  two  points,  as  e 
and  fy  and  describe  arcs  intersecting,  as  at  g  and  h  ;  join  g 
and  h  and  c  and  d\  the  intersection  o  is  the  centre. 

This  is  upon  the  same  principle  as  Art.  514. 

A  third  method.  Draw  any  chord,  as  ab  (Fig.  363),  and 
from  the  point  a  draw  ac  at  right  angles  to  ab  ;  join  c  and 
b\  bisect  c  b  at  d — which  will  be  the  centre  of  the  circle. 


A   TANGENT  AT  A  GIVEN   POINT. 


557 


If  a  circle  be  not  too  large  for  the  purpose,  its  centre 
may  very  readily  be  ascertained  by  the  help  of  a  carpenters'- 
square,  thus :  apply  the  corner  of  the  square  to  any  point  in 
the  circumference,  as  at  a ;  by  the  edges  of  the  square 
(which  the  lines  ab  and  ac  represent)  draw  lines  cutting  the 


FIG.  363. 

circle,  as  at  b  and  c ;  join  b  and  c ;  then  if  be  is  bisected,  as  at 
d,  the  point  d  will  be  the  centre.     (See  Art.  352.) 

5 II  „ — At  a  Given   Point  in  a  Circle  to  Draw  a  Tangent 
thereto. — Let  a  (Fig.  364)  be  the  given  point,  and  b  the  cen- 


FIG.  364. 

tre  of  the  circle.     Join  a  and  b ;  through  the  point  a,  and  at 
right  angles  to  a  b,  draw  cd ;  then  c  d  is  the  tangent  required. 

512. — The  Same,  without  making  use  of  the  Centre  of 
the  Circle. — Let  a  (Fig.  365)  be  the  given  point.  From  a  set 
off  any  distance  to  b,  and  the  same  from  b  to  c ;  join  a  and 
c  ;  upon  a,  with  ab  for  radius,  describe  the  arc  dbc-,  make 
db  equal  to  bc\  through  a  and  d  draw  a  line;  this  will  be 
the  tangent  required. 


558  PRACTICAL   GEOMETRY. 

The  correctness  of  this  method  depends  upon  the  fact 
that  the  angle  formed  by  a  chord  and  tangent  is  equal  to  any 
inscribed  angle  in  the  opposite  segment  of  the  circle  (Art. 
358);  ab  being  the  chord,  and  bca  the  angle  in  the  opposite 
segment  of  the  circle.  Now,  the  angles  dab  and  bca  are 
equal,  because  the  angles  dab  and  bac  are,  by  construction, 


FIG.  365. 

equal;  and  the  angles  bac  and  bca  are  equal,  because  the 
triangle  abc  is  an  isosceles  triangle,  having  its  two  sides,  ab 
and  be,  by  construction  equal ;  therefore  the  angles  dab  and 
bca  are  equal. 

513. — A  Circle  and  a  Tangent  Given,  to  Find  Hie  Point 
of  Contact.— From  any  point,  as  a  (Fig.  366),  in  the  tangent 


FIG.  366. 

be,  draw  a  line  to  the  centre  d\  bisect  ad  at  r;  upon  e,  with 
the  radius  ea,  describe  the  arc  afd\  f  is  the  point  of  con- 
tact required. 

If/  and//  were  joined,  the  line  would  form  right  angles 
with  the  tangent  be.     (See  Art.  352.) 


A   CIRCLE   THROUGH   GIVEN   POINTS. 


559 


514. — Through  any  Three  Points  not  in  a  Straight  Line, 
to  Draw  a  Circle. — Let  a,  b  and  c  (Fig.  367)  be  the  three 
given  points.  Upon  a  and  b,  with  any  radius  greater  than 
half  a  b,  describe  arcs  intersecting  at  d  and  e ;  upon  b  and  c, 
with  any  radius  greater  than  half  be,  describe  arcs  intersect- 
ing at  /  and  g\  through  d  and  e  draw  a  right  line,  also 


..      FIG.  367. 

another  through  /  and  g\  upon  the  intersection  //,  with  the 
radius  ha,  describe  the  circle  a  be,  and  it  will  be  the  one  re- 
quired. 


515. — Three  Points  not  in  a  Straight  Line  being  Given, 
to  Find  a  Fourth  that  shall,  with  the  Three,  Lie  in  the 
Circumference  of  a  Circle. — Let  a  b  c  (Fig.  368)  be  the  given 
points.  Connect  them  with  right  lines,  forming  the  triangle 


FIG.  368. 

acb',  bisect  the  angle  cba  (Art.  506)  with  the  line  bd\  also 
bisect  c  a  in  e,  and  erect  ed  perpendicular  to  ac,  cutting  bd 
in  d;  then  d  is  the  fourth  point  required. 

A  fifth  point  may  be  found,  as  at  /,  by  assuming  a,  d  and 
b,  as  the  three  given  points,  and  proceeding  as  before.     So, 


560  PRACTICAL   GEOMETRY. 

also,  any  number  of  points  may  be  found  simply  by  using 
any  three  already  found.  This  problem  will  be  serviceable 
in  obtaining  short  pieces  of  very  flat  sweeps.  (See  Art.  240.) 
The  proof  of  the  correctness  of  this  method  is  found  in 
the  fact  that  equal  chords  subtend  equal  angles  (Art.  357). 
Join  d  and  c;  then  since  ae  and  ec  are,  by  construction, 
equal,  therefore  the  chords  a  d  and  dc  are  equal ;  hence  the 
angles  they  subtend,  dba  and  d b  c,  are  equal.  So,  like- 
wise, chords  drawn  from  a  to  /,  and  from  /  to  d,  are  equal, 
and  subtend  the  equal  angles  dbf  and  fba.  Additional 
points  beyond  a  or  b  may  be  obtained  on  the  same  principle. 
To  obtain  a  point  beyond  a,  on  b,  as  a  centre,  describe  with 
any  radius  the  arc  ion ;  make  on  equal  to  o  i ;  through  b  and 
n  draw  b g\  on  a  as  centre  and  with  af  for  radius,  describe 
the  arc,  cutting  gb  at  gt  then  g.  is  the  point  sought. 

516. — To  Describe  a  Segment  of  a  Circle  toy  a  Set-Tri- 
angle.— Let  a  b  (Fig.  369)  be  the  chord,  and  c  d  the  height 


FIG.  369. 

of  the  segment.  Secure  two  straight-edges,  or  rulers,  in  the 
position  ce  and  c  f,  by  nailing  them  together  at  c,  and  affixing 
a  brace  from  c  to  /;  put  in  pins  at  a  and  b  ;  move  the  angu- 
lar point  c  in  the  direction  acb\  keeping  the  edges  of  the 
triangle  hard  against  the  pins  a  and  b ;  a  pencil  held  at  c 
will  describe  the  arc  acb. 

A  curve  described  by  this  process  is  accurately  circular, 
and  is  not  a  mere  approximation  to  a  circular  arc,  as  some 
may  suppose.  This  method  produces  a  circular  curve,  be- 
cause all  inscribed  angles  on  one  side  of  a  chord-line  are 
equal  (Art.  356).  To  obtain  the  radius  from  a  chord  and  its 
versed  sine,  see  Art.  444. 

If  the  angle  formed  by  the  rulers  at  c  be  a  right  angle, 


TO   FIND   THE  VERSED   SINE.  561 

the  segment  described  will  be  a  semi-circle.  This  problem 
is  useful  in  describing  centres  for  brick  arches,  when  they 
are  required  to  be  rather  flat.  Also,  for  the  head  hang- 
ing-stile of  a  window-frame,  where  a  brick  arch,  instead  of  a 
stone  lintel,  is  to  be  placed  over  it. 

517. — To  Find  the  Radiu§  of  an  Arc  of  a  Circle  when 
the  Chord  and  Ver§ed  Sine  are  Given. — The  radius  is  equal 
to  the  sum  of  the  squares  of  half  the  chord  and  of  the  versed 
sine,  divided  by  twice  the  versed  sine.  This  is  expressed, 

(-}*        a 
algebraically,  thus  :  r  =  —        — ,  where  r  is  the  radius,  c  the 

chord,  and  v  the  versed  sine  (Art.  444). 

Example. — In  a  given  arc  of  a  circle  a  chord  of  12  feet 
has  the  rise  at  the  middle,  or  the  versed  sine,  equal  to  2  feet, 
what  is  the  radius  ? 

Half  the  chord  equals  6,  the  square  of  6  is,  6  x  6  =  36 
The  square  of  the  versed  sine  is,  2x2=4 

Their  sum  equals,  40 

Twice  the  versed  sine  equals  4,  and  40  divided  by  4  equals 
10.  Therefore  the  radius,  in  this  case,  is  10  feet.  This 
result  is  shown  in  less  space  and  more  neatly  by  using  the 
above  algebraical  formula.  For  the  letters  substituting 

their  value,  the  formula  r  =  — — i- —  becomes  r  =  sj£ 

2V  2X2 

and  performing  the  arithmetical  operations  here  indicated 
equals — 

6 a  -f  2 2   _  36  -t-  4  _  40  _ 

4  44" 

518. — To  Find  the  Ver§ed  Sine  of  an  Arc  of  a  Circle 
when  the  Radius  and  Chord  are  Given. — The  versed  sine 
is  equal  to  the  radius,  less  the  square  root  of  the  difference 
of  the  squares  of  the  radius  and  half  chord  ;  expressed  alge- 
braically thus :  v  =  r  —  Vr 3  -  (l)a,  where  r  is  the  radius,  v 
the  versed  sine,  and  c  the  chord.  (Equation  (161.)  reduced.) 


562  PRACTICAL   GEOMETRY. 

Example. — In  an  arc  of  a  circle  whose  radius  is  75  feet, 
what  is  the  versed  sine  to  a  chord  of  120  feet?  By  the  table 
in  the  Appendix  it  will  be  seen  that — 

The  square  of  the  radius,  75,  equals  .         .  5625 
The  square  of  half  the  chord,  60,  equals  .  5600 

The  .difference  is          .....  2025 

The  square  root  of  this  is   .         .         .         -45 

This  deducted  from  the  radius  ...       75 

• 
The  remainder  is  the  versed  sine,  =  30 

This  is  expressed  by  the  formula,  thus — 


v  =  75  -  ^75  "  -  FF?  =  75  -  ^5625-3600  =  75  -  45  =  30. 

519. — To  Describe  the  Segment  of  a  Circle  by  Intersec- 
tion of  Lines. — Let  ab  (Fig.  370)  be  the  chord,  and  cd  the 


height  of  the  segment.  Through  c  draw  ef  parallel  to  a  b  ; 
draw  bf  at  right  angles  to  cb;  make  ce  equal  to  cf;  draw 
ag  and  bh  at  right  angles  to  a  b  ;  divide  ce,  cft  da,  db,  a g, 
and  bh,  each  into  a  like  number  of  equal  parts,  as  four; 
draw  the  lines  i  1,22,  etc.,  and  from  the  points  o,  o,  and  o, 
draw  lines  to  c\  at  the  intersection  of  these  lines  trace  the 
curve,  acb,  which  will  be  the  segment  required. 

In  very  large  work,  or  in  laying  out  ornamental  gar- 
dens, etc.,  this  will  be  found  useful ;  and  where  the  centre 
of  the  proposed  arc  of  a  circle  is  inaccessible  it  will  be  inval- 
uable. (To  trace  the  curve,  see  note  at  Art.  550.) 

The  lines  e  a,  c  d,  and  fb,  would,  were  they  extended, 
meet  in  a  point,  and  that  point  would  be  in  the  opposite 
side  of  the  circumference  of  the  circle  of  which  acb  is  a 


ORDINATES   TO   AN  ARC. 


563 


segment.  The  lines  i  i,  2  2,  3  3,  would  likewise,  if  extended, 
meet  in  the  same  point.  The  line  cd,  if  extended  to  the  op- 
posite side  of  the  circle,  would  become  a  diameter.  The  line 
fb  forms,  by  construction,  a  right  angle  with  be,  and  hence 
the  extension  of  fb  would  also  form  a  right  angle  with  be, 
on  the  opposite  side  of  bc\  and  this  right  angle  would  be 
the  inscribed  angle  in  the  semi-circle ;  and  since  this  is  re- 
quired to  be  a  right  angle  (Art.  352),  therefore  the  construc- 
tion thus  far  is  correct,  and  it  will  be  found  likewise  that  at 
each  point  in  the  curve  formed  by  the  intersection  of  the 
radiating  lines,  these  intersecting  lines  are  at  right  angles. 

520.  —  Ordinate§.  —  Points  in  the  circumference  of  a 
circle  may  be  obtained  arithmetically,  and  positively  accu- 
rate, by  the  calculation  of  ordinates,  or  the  parallel  lines  o  i, 


«  -f  J  2  /  (1725 

FIG.  371. 

02,  03,  04  (Fig.  3/i).  These  ordinates  are  drawn  at  right 
angles  to  the  chord-line  a  b,  and  they  may  be  drawn  at  any 
distance  apart,  either  equally  distant  or  unequally,  and  there 
may  be  as  many  of  them  as  is  desirable ;  the  more  there  are 
the  more  points  in  the  curve  will  be  obtained.  If  they  are 
located  in  pairs,  equally  distant  from  the  versed  sine  c  d, 
calculation  need  be  made  only  for  those  on  one  side  of  cd, 
as  those  on  the  opposite  side  will  be  of  equal  lengths,  re- 
spectively;  for  example:  o  i,  on  the  left-hand  side  of  cd,  is 
equal  to  o  i  on  the  right-hand  side,  o  2  on  the  right  equals 
o  2  on  the  left,  and  in  like  manner  for  the  others. 

The  length  of  any  ordinate  is  equal  to  the  square  root 
of  the  difference  of  the  squares  of  the  radius  and  abscissa, 
less  the  difference  between  the  radius  and  versed  sine  (Art. 
445).  The  abscissa  being  the  distance  from  the  foot  of 
the  versed  sine  to  the  foot  of  the  ordinate.  Algebraically, 


564  PRACTICAL   GEOMETRY. 


t  =  Vr  *  •-  x*  —  (r  —  b\  where  t  is  put  to  represent  the  ordi- 
nate  ;  x,  the  abscissa  ;  b,  the  versed  sine  ;  and  r,  the  radius. 

Example.  —  An  arc  of  a  circle  has  its  chord  ab  (Fig.  371) 
100  feet  long,  and  its  versed  sine  cd,  5  feet.  It  is  required 
to  ascertain  the  length  of  ordinates  for  a  sufficient  number 
of  points  through  which  to  describe  the  curve.  To  this  end 
it  is  requisite,  first,  to  ascertain  the  radius.  This  is  readily 

/£\2  2 

done  in  accordance  with  Art.  517.      For  -  becomes 

-  =  252-5  =  radius.     Having  the  radius,  the  curve 

2x5 

might  at  once  be  described  without  the  ordinate  points,  but 
for  the  impracticability  that  usually  occurs,  in  large,  flat 
segments  of  the  circle,  of  getting  a  location  for  the  centre, 
the  centre  usually  being  inaccessible.  The  ordinates  are, 
therefore,  to  be  calculated.  In  Fig.  371  the  ordinates  are 
located  equidistant,  and  are  10  feet  apart.  It  will  only 
be  requisite,  therefore,  to  calculate  those  on  one  side  of 
the  versed  sine  cd.  For  the  first  ordinate  01,  the  formula 
/  =  Vr^—  x*  —  (r  —  &)  becomes— 

2-  io2-(252-5  -5). 


=  1/63756-25  —  100  —  247.5. 

252.3019-247.5. 

4.8019  =  the  first  ordinate,  o  i. 
For  the  second  — 

t—  ^252-  52  —  202  —  (252-5  —  5). 
=     251-7066  —  247.5. 

4-2066  =  the  second  ordinate,  02. 
For  the  third— 

/  =  1/^52-5  a-3o2  -  247.  5. 
=     250-7115-247.5. 

3-2115  =  the  third  ordinate,  03. 


TO   DESCRIBE  A   TANGED   CURVE.  565 

For  the  fourth— 

/  =  1/252. 5  2  -402  -  247-  5. 
=     249-3115  -  247.5. 

1-8115  =  the  fourth  ordinate,  o 4. 

The  results  here  obtained  are  in  feet  and  decimals  of  a 
foot.  To  reduce  these  to  feet,  inches,  and  eighths  of  an 
inch,  proceed  as  at  Reduction  of  Decimals  in  the  Appendix. 
If  the  two-feet  rule,  used  by  carpenters  and  others,  were 
decimally  divided,  there  would  be  no  necessity  of  this  re- 
duction, and  it  is  to  be  hoped  that  the  rule  will  yet  be  thus 
divided,  as  such  a  reform  would  much  lessen  the  labor  of 
computations,  and  insure  more  accurate  measurements. 

Versed  sine  c  d  =  ft.  5  «o         =  ft.  5  -o  inches. 
Ordinates     o  I  =       4-8019=       4-9!  inches,  nearly. 

"  02=       4-2066  =       4. 2\  inches,  nearly. 

"  03=       3-2115   —       3 -2j  inches,  nearly. 

"  04=       1-8115   =       I- 9f  inches,  nearly. 

521. — In  a  Given  Angle,  to  Describe  a  Tanged  Curve. 

— Let  a  b  c  (Fig.  372)  be  the  given  angle,  and  I  in  the  line  a  b, 


FIG.  372. 

and  5  in  the  line  be,  the  termination  of  the  curve.  Divide 
i  b  and  b  5  into  a  like  number  of  equal  parts,  as  at  i,  2,  3,  4, 
and  5  ;  join  i  and  i,  2  and  2,  3  and  3,  etc. ;  and  a  regular 
curve  wiU  be  formed  that  will  be  tangical  to  the  line  ab,  at 
the  point  i,  and  to  be  at  5. 

This  is  of  much  use  in  stair-building,  in  easing  the  angles 
formed  between  the  wall-string  and  the  base  of  the  hall, 
also  between  the  front  string  and  level  facia,  and  in  many 
other  instances.  The  curve  is  not  circular,  but  of  the  form 
of  the  parabola  (Fig.  418) ;  yet  in  large  angles  the  difference 


PRACTICAL   GEOMETRY. 


is  not  perceptible.  This  problem  can  be  applied  to  describ- 
ing- the  curve  for  door-heads,  window-heads,  etc.,  to  rather 
better  advantage  .than  Art.  516.  For  instance,  let  ab  (Fig. 
373)  be  the  width  of  the  opening,  and  c  d  the  height  of  the 


«  c  b 

FIG.  373. 

arc.     Extend  c  d,  and   make  de  equal  to  cd\  join  a  and  e, 
also  e  and  b ;  and  proceed  as  directed  above. 

522. — To  De§eribe  a  Circle  within  any  Given  Triangle, 
so    that   the    Sides  of  the  Triangle   shall   be   Tangical. — 

Let  a  be  (Fig.  374)  be  the  given  triangle.     Bisect  the  angles 


FIG.  374. 

a  and  b  according  to  Art.  506;  upon  d,  the  point  of  intersec- 
tion of  the  bisecting  lines,  with  the  radius  d  e,  describe  the 
required  circle. 

523. — About  a  Given  Circle,  to  Describe  an  Equilateral 
Triangle. — Let  adb c  (Fig.  375)  be  the  given  circle.  Draw 
the  diameter  c  d;  upon'af,  with  the  radius  of  the  given  circle, 
describe  the  arc  aeb\  join  a  and  b ;  draw  fg  at  right  angles 
to  dc ;  make  fc  and  eg  each  equal  to  ab ;  from  ft  through 
a,  draw  //*,  also  from  g,  through  b,  draw  gh\  then  fgh 
will  be  the  triangle  required. 

524. — To   Find  a  Right  L.ine  nearly  Equal  to  the  Cir- 
cumference of  a  Circle. — Let  abed  (Fig.  376)  be  the  given 


A   RIGHT   LINE   EQUAL  TO  A  CIRCUMFERENCE.          567 

circle.  Draw  the  diameter  ac\  on  this  erect  an  equilateral 
triangle  aec  according  to  Art.  525  ;  draw  gf  parallel  to  ac\ 
extend  ec  to  /,  also  ea  to  g\  then  gf  will  be  nearly  the 


*   ^  FIG.  375. 

length  of  the  semi-circle  adc\  and  twice  gf  will  nearly 
equal  the  circumference  of  the  circle  a  b  c  d,  as  was  required. 
Lines  drawn  from  *,  through  any  points  in  the  circle,  as 
o,  o  and  o,  to  /,/  and  /,  will  divide  gf  in  the  same  way  as 
the  semi-circle  adc  is  divided.  So,  any  portion  of  a  circle 
may  be  transferred  to  a  straight  line.  This  is  a  very  useful 


ff      P      P  d  p  f 

FIG.  376. 


problem,  and   should   be  well  studied,   as  it  is  frequently 
used  to  solve  problems  on  stairs,  domes,  etc. 

Another  method.     Let  a  bfc  (Fig.  377)  be  the  given  circle. 
Draw  the  diameter  ac\  from  d,  the  centre,  and  at  right  an- 


568  PRACTICAL   GEOMETRY. 

gles  to  ac,  draw  db\  join  b  and  c\  bisect  be  at  e\  from  d, 
through  e,  draw  df\  then  ef  added  to  three  times  the  di- 
ameter, will  equal  the  circumference  of  the  circle  sufficiently 
near  for  many  uses.  The  result  is  a  trifle  too  large.  If  the 


FIG.  377. 

circumference  found  by  this  rule  be  divided  by  648-22  the 
quotient  will  be  the  excess.  Deduct  this  excess,  and  the 
remainder  will  be  the  true  circumference.  This  problem  is 
rather  more  curious  than  useful,  as  it  is  less  labor  to  perform 
the  operation  arithmetically,  simply  multiplying  the  given 
diameter  by  3- 1416,  or,  where  a  greater  degree  of  accuracy 
is  needed,  by  3-1415926.  (See  Art.  446.) 

POLYGONS,    ETC. 

525. — Upon  a  Given  Line  to  Construct  an  Equilateral 
Triangle. — Let  a  b  (Fig.  378)  be  the  given  line.     Upon  a  and 


« 

FIG.  378. 


b,  with  a  b  for  radius,  describe  arcs,  intersecting  at  c  ;  join  a 
and  c,  also  c  and  b  ;  then  acb  will  be  the  triangle  required. 

526.  —  To  De§cribe  an  Equilateral  Rectangle,  or  Square. 

—  Let  a  b  (Fig.  379)  be  the  length  of  a  side  of  the  proposed 


POLYGONS   IN   CIRCUMSCRIBING   CIRCLES. 


square.  Upon  a  and  b,  with  a  b  for  radius,  describe  the  arcs 
a  d  and  b  c  ;  bisect  the  arc  ae  in  f.\  upon  e,  with  ef  for  ra- 
dius, describe  the  arc  cfd\  join  a  and  <:,  c  and  </,  </and  £; 
then  acdb  will  be  the  square  required. 


FIG.  379. 

527. — Within  a  Given  Circle,  to  Inscribe  an  Equilateral 
Triangle,  Hexagon  or  Dodecagon. — Let  abed  (Fig.  380)  be 
the  given  circle.  Draw  the  diameter  bd\  upon  b,  with  the 
radius  of  the  given  circle,  describe  the  arc  ae  c ;  join  a  and  c, 
also  a  and  </,  and  c  and  </— and  the  triangle  is  completed. 
For  the  hexagon:  from  a,  also  from  c,  through  e,  draw  the 
lines  af  and  cg\  join  #  and  b,  b  and  <:,  c  and  /,  etc.,  and  the 


hexagon  is  completed.     The  dodecagon  may  be  formed  by 
bisecting  the  sides  of  the  hexagon. 

Each  side  of  a  regular  hexagon  is  exactly  equal  to  the 
radius  of  the  circle  that  circumscribes  the  figure.  For  the 
radius  is  equal  to  a  chord  of  an  arc  of  60  degrees ;  and,  as 
every  circle  is  supposed  to  be  divided  into  360  degrees,  there 
is  just  6  times  60,  or  6  arcs  of  60  degrees,  in  the  whole  cir- 
cumference. A  line  drawn  from  each  angle  of  the  hexagon 


570 


PRACTICAL   GEOMETRY. 


to  the  centre  (as  in  the  figure)  divides  it  into  six  equal,  equi- 
lateral triangles. 

528. — Within  a  Square  to  Inscribe  an  Octagon. — Let 

abed  (Fig.  381)  be  the  given  square.     Draw  the  diagonals 


a  d  and  b  c ;  upon  a,  b,  c,  and  d,  with  a  e  for  radius,  describe 
arcs  cutting  the  sides  of  the  square  at  1,2,  3,  4,  5,  6,  7,  and  8  ; 
join  i  and  2,  3  and  4,  5  and  6,  etc.,  and  the  figure  is  com- 
pleted. 

In  order  to  eight-square  a  hand-rail,  or  any  piece  that  is 


FIG.  382. 


to  be  afterwards  rounded,  draw  the  diagonals  a  d  and  b  c 
upon  the  end  of  it,  after  it  has  been  squared-up.  Set  a 
gauge  to  the  distance  ae  and  run  it  upon  the  whole  length 
of  the  stuff,  from  each  corner  both  ways.  This  will  show 


BUTTRESSED   OCTAGON.  5/1 

how  much  is  to  be  chamfered  off,  in  order  to  make  the  piece 
octagonal  (Art.  354). 

529. — To  Find  the  Side  of  a  Buttressed  Octagon. — Let 

ABCDE  (Fig.  382)  represent  one  quarter  of  an  octagon 
structure,  having  a  buttress  H FGJ  at  each  angle.  The 
distance  M H,  between  the  buttresses,  being  given,  as  also 
F  G,  the  width  of  a  buttress ;  to  find  H  C  or  C  J,  in  order  to 
obtain  B  C,  the  side  of  the  octagon.  Let  B  C,  a  side  of  the 
octagon,  be  represented  by  b ;  or  D  C  by  £  b.  Let  M  H  =  a  ; 
or  J D  =  ±a\  and 
Then  we  have — 


JD+  JC  =  CD, 

.  i  a  +  x  =  i  b, 

a  +  2  x  =  b. 

For  FG  put  p\  ^r  LG  —  K  J  —  \p. 

Now  £  D  is  the  radius  of  an  inscribed  circle  and,  as  per 

equation  (140.),  equals  r  —  (  1/2  +  i)  -. 

Also,  £  (7  is  the  radius  of  a  circumscribed  circle,  and,  as 
per  equation  (141.)*  equals  R—  ^2^2  +  4-. 

The  two  triangles,  CJK  and  C  ED,  are  homologous; 
for  the  angles  at  C  are  common  and  the  angles  at  AT  and  D 
are  right  angles.  Having  thus  two  angles  of  one  equal 
respectively  to  the  two  angles  of  the  other,  therefore  (Art. 
345)  the  remaining  angles  must  be  equal.  Hence,  the  sides 
of  the  triangles  are  proportionate,  or— 

ED  :  EC  ::  JK  :  CJ 
r  •  R  '.'.  \} 

The  value  of  the  side,  as  above,  is— 

R 

I  =  a  +  2  x  =  a  +  p  -  - , 


572  pkACTICAL  GEOMETRY. 

And  taking  the  value  of  R  and  r,  as  above,  we  have — 


(1/2 


+1 


r> 

Substituting  this  for  — ,  we  have — 


V2  +  I 

The  numerical  coefficient  of  /  reduces  to  1-0823923  or 
i  -0824,  nearly. 

Therefore  we  have — 

b  =  a  +  i  -o824/.  (207.) 

Or:  The  side  of  a  buttressed  octagon  equals  the  distance  be- 
tween the  buttresses  plus  \  -0824  times  the  width  of  the  faced 
the  buttress. 

For  example  :  let  there  be  an  octagon  building,  which 
measures  between  the  buttresses,  as  at  M H,  18  feet,  and  the 
face  of  the  buttresses,  as  FG,  equals  3  feet ;  what,  in  such  a 
building,  is  the  length  of  a  side  B  Cl  For  this,  using  equa- 
tion (207.),  we  have — 

b  =  1 8  +  I  -0824  x  3 
=  18  +  3-2472 
=  21-2472. 

Or :  The  side  of  the  octagon  B  C  equals  21  feet  and  nearly  3 
inches. 

530. — Within  a  Given  Circle  to  In§cribe  any  Regular 
Polygon. — Let  abc  2  (Figs.  383*  384,  and  385)  be  given  circles. 
Draw  the  diameter  a  c ;  upon  this  erect  an  equilateral  trian- 
gle aec,  according  to  A rt.  525  ;  divide  ac  into  as  many  equal 
parts  as  the  polygon  is  to  have  sides,  as  at  i,  2,  3,  4,  etc.; 
from  e,  through  each  even  number,  as  2,  4,  6,  etc.,  draw  lines 


TO  DESCRIBE  ANY  REGULAR  POLYGON. 


573 


cutting  the  circle  in  the  points  2,  4,  etc.;  from  these  points 
and  at  right  angles  to  a  c  draw  lines  to  the  opposite  part 
of  the  circle ;  this  will  give  the  remaining  points  for  the 
polygon,  as  b,  /,  etc. 

In  forming  a  hexagon,  the  sides  of  the  triangle  erected 


FIG.  383. 


upon  ac  (as  at  Fig.  384)  mark  the  points  b  and  f.  This 
method  of  locating  the  angles  of  a  polygon  is  an  approxima- 
tion sufficiently  near  for  many  purposes ;  it  is  based  upon 
the  like  principle  with  the  method  of  obtaining  a  right  line 
nearly  equal  to  a  circle  (Art.  524).  The  method  shown  at 
Art.  531  is  accurate. 


FIG.  386. 


FIG.  387.- 


FIG.  388 


531.— Upon  a  Given  Line  to  De§crifoe  any  Regular 
Polygon.— Let  a  b  (Figs.  386,  387,  and  388)  be  (riven  lines, 
equal  to  a  side  of  the  required  figure.  From  b  draw  be  at 
right  angles  to  a  b  ;  upon  a  and  &,  with  a  b  for  radius,  describe 
the  arcs  acd  and  feb\  divide  ac  into  as  many  equal  parts 


5/4  PRACTICAL   GEOMETRY. 

as  the  polygon  is  to  have  sides,  and  extend  those  divisions 
from  c  towards  d\  from  the  second  point  of  division,  count- 
ing from  c  towards  a,  as  3  (Fig.  386),  4  (Fig.  387),  and  5  (Fig. 
388),  draw  a  line  to  b ;  take  the  distance  from  said  point  of 
division  to  a,  and  set  it  from  b  to  e ;  join  e  and  a ;  upon  the 
intersection  o  with  the 'radius  oa,  describe  the  circle  afdb; 
then  radiating  lines,  drawn  from  b  through  the  even  numbers 
on  the  arc  a  d,  will  cut  the  circle  at  the  several  angles  of  the 
required  figure. 

In  the  hexagon  (Fig.  387),  the  divisions  on  the  arc  ad  are 
not  necessary ;  for  the  point  o  is  at  the  intersection  of  the 
arcs  ad  and  fb,  the  points  f  and  d  are  determined  by  the 
intersection  of  those  arcs  with  the  circle,  and  the  points 
above  g  and  h  can  be  found  by  drawing  lines  from  a  and  b 
through  the  centre  o.  In  polygons  of  a  greater  number  of 
sides  than  the  hexagon  the  intersection  o  comes  above  the 
arcs ;  in  such  case,  therefore,  the  lines  a  e  and  b  5  (Fig.  388) 
have  to  be  extended  before  they  will  intersect.  This  method 
of  describing  polygons  is  founded  on  correct  principles,  and 
is  therefore  accurate.  In  the  circle  equal  arcs  subtend 
equal  angles  (Arts.  357  and  515).  Although  this  method  is 
accurate,  yet  polygons  may  be  described  as  accurately  and 
more  simply  in  the  following  manner.  It  will  be  observed 
that  much  of  the  process  in  this  method  is  for  the  purpose 
of  ascertaining  the  centre  of  a  circle  that  will  circumscribe 
the  proposed  polygon.  By  reference  to  the  Table  of  Poly- 
gons in  Art.  442  it  will  be  seen  ho-w  this  centre  may  be  ob- 
tained arithmetically.  This  is  the  rule  :  multiply  the  given 
side  by  the  tabular  radius  for  polygons  of  a  like  number  of 
sides  with  the  proposed  figure,  and  the  product  will  be  the 
radius  of  the  required  circumscribing  circle.  Divide  this 
circle  into  as  many  equal  parts  as  the  polygon  is  to  have 
sides,  connect  the  points  of  division  by  straight  lines,  and 
the  figure  is  complete.  For  example :  It  is  desired  to  de- 
scribe a  polygon  of  7  sides,  and  20  inches  a  side.  The  tabu- 
lar radius  is  1-15238.  This  multiplied  by  20,  the  product, 
23-0476  is  Ihe  required  radius  in  inches.  The  Rules  for 
the  Reduction  of  Decimals,  in  the  Appendix,  show  how  to 
change  decimals  to  the  fractions  of  a  foot  or  an  inch.  From 


EQUAL  FIGURES. 


575 


this,  23  -0476  is  equal  to  23TV  inches,  nearly.  It  is  not  needed 
to  take  all  the  decimals  in  the  table,  three  or  four  of  them 
will  give  a  result  sufficiently  near  for  all  ordinary  practice. 

532. — To  €on§truct  a  Triangle  whose  Side§  shall  be 
§everally  Equal  to  Three  Given  Lines. — Let  a,  b  and  c  (Fig. 
389)  be  the  given  lines.  Draw  the  line  de  and  make  it  equal 


FIG.  389. 

c\  upon  e,  with  b  for  radius,  describe  an  arc  at  /;  upon  d, 
with  a  for  radius,  describe  an  arc  intersecting  the  other  at/; 
join  d  and  /,  also  f  and  e ;  then  dfe  will  be  the  triangle 
required. 

533 — To  Construct  a  Figure  Equal  to  a  Given,  Right- 
lined  Figure.  —  Let  abed  (Fig.  390)  be  the  given  figure. 
Make  ef  (Fig.  391)  equal  to  cd-,  upon  /,  with  da  for  radius, 


FIG.  390. 


FIG.  391. 


describe  an  arc  at^;  upon  r,  with  ca  for  radius,  describe  an 
arc  intersecting  the  other  at^;  join  ^and  e,\  upon  f  and  g, 
with  db  and  ab  for  radius,  describe  arcs  intersecting  at  // ; 
join  g  and  /*,  also  h  and  /;  then  Fig.  391  will  every  way 
equal  Fig.  390. 

So,  right-lined  figures  of  any  number  of  sides  may  be 
copied,  by  first  dividing  them  into  triangles,  and  then  pro- 


576 


PRACTICAL   GEOMETRY. 


ceeding  as  above.  The  shape  of  the  floor  of  any  room,  or 
of  any  piece  of  land,  etc.,  may  be  accurately  laid  out  by  this 
problem,  at  a  scale  upon  paper  ;  and  the  contents  in  square 
feet  be  ascertained  by  the  next. 

534. — To  Make  a  Parallelogram  equal  to  a  Given 
Triangle. — Let  a  be  (Fig.  392)  be  the  given  triangle.  From 
a  draw  a  d  at  right  angles  to  b  c ;  bisect  a  d  in  e ;  through  e 


f 


FIG.  392. 

draw  fg  parallel  to  be-,  from  b  and  c  draw  b  f  and  eg ^par- 
allel to  de\  then  bfgc  will  be  a  parallelogram  containing  a 
surface  exactly  equal  to  that  of  the  triangle  a  be. 

Unless  the  parallelogram  is  required  to  be  a  rectangle, 
the  lines  bf  and  eg  need  not  be  drawn  parallel  to  d  e.  If  a 
rhomboid  is  desired  they  may  be  drawn  at  an  oblique  angle, 
provided  they  be  parallel  to  one  another.  To  ascertain  the 
area  of  a  triangle,  multiply  the  base  be  by  half  the  perpen- 


d 
FIG.  393. 


dicular  height  da. 
is  taken  for  base. 


In  doing  this  it  matters  not  which  side 


535. — A  Parallelogram  being  Given,  to  Construct  An- 
other Equal  to  it,  and  Having  a  Side  Equal  to  a  Given  Line. 

—Let  A  (Fig.  393)  be  the  given  parallelogram,  and  B  the 
given  line.     Produce  the  sides  of  the  parallelogram,  as  at 


SQUARE   EQUAL  TO   TWO   OR   MORE   SQUARES.  577 

a,  b,  c,  and  d'\  make  ed  equal  to  B ;  through  d  draw  cf  par- 
allel to  gb-,  through  e  draw  the  diagonal  ca\  from  a  draw 
af  parallel  to  ed;  then  C  will  be  equal  to  A.  (See  Art.  340.) 

536. — To  Make  a  Square  Equal  to  two  or  more  Given 
Squares. — Let  A  and  B  (Fig.  394)  be  two  given  squares. 


FIG.  394. 

Place  them  so  as  to  form  a  right  angle,  as  at  a  ;  Join  b  ai<d  c ; 
then  the  square  C,  formed  upon  the  line  be,  wi'J1  be  equal  in 
extent  to  the  squares  A  and  B  added  together.  Again :  if 
a  b  (Fig.  395)  be  equal  to  the  side  of  a  given  square,  c  a,  placed 
at  right  angles  to  a  b,  be  the  side  of  another  given  square, 


and  cd,  placed  at  right  angles  to  cb,  be  the  side  of  a  third 
given  square,  then  the  square  A,  formed  upon  the  line  db> 
will  be  equal  to  the  three  given  squares.  (See  Art.  353.) 

The  usefulness  and  importance  of  this  problem  are  pro- 
verbial.    To  ascertain  the  length  of  braces  and  of  rafters  in 


578  PRACTICAL   GEOMETRY. 

framing,  the  length  of  stair-strings,  etc.,  are  some  of  the  pur- 
poses to  which  it  may  be  applied  in  carpentry.  (See  note 
to  Art.  503.)  If  the  lengths  of  any  two  sides  of  a  right- 
angled  triangle  are  known,  that  of  the  third  can  be  ascer- 
tained. Because  the  square  of  the  hypothenuse  is  equal  to 
the  united  squares  of  the  two  sides  that  contain  the  right 
angle. 

(i.) — The  two  sides  containing  the  right  angle  being 
known,  to  find  the  hypothenuse. 

Rule. — Square  each  given  side,  add  the  squares  together, 
and  from  the  product  extract  the  square  root ;  this  will  be 
the  answer. 

For  instance,  suppose  it  were  required  to  find  the  length 
of  a  rafter  for  a  house,  34  feet  wide — the  ridge  of  the  roof 
to  be  9  feet  high,  above  the  level  of  the  wall-plates.  Then 
17  feet,  half  of  the  span,  is  one,  and  9  feet,  the  height,  is  the 
other  of  the  sides  that  contain  the  right  angle.  Proceed  as 
directed  by  the  rule : 

17  9 

17  _9 

119  8 1  =  square  of  9. 

17  289  =  square  of  17. 

289  —  square  of  17.     370  Product. 

i  )  370  (  19-235  +  =  square  root  of  370 ;  equal  19  feet  2-J  in., 
i  i  nearly  ;  which  would  be  the  required 

20  )~270  length  of  the  rafter. 

9    261 
382).  -900 

_2      ^1 
3843)  13600 


38465)- 207 100 
192325 


TO    FIND   THE    LENGTH    OF  A   RAFTER.  579 

(By  reference  to  the  table  of  square  roots  in  the  Appen- 
dix, the  root  of  almost  any  number  may  be  found  ready 
calculated  ;  also,  to  change  the  decimals  of  a  foot  to  inches 
and  parts,  see  Rules  for  the  Reduction  of  Decimals  in  the 
Appendix.) 

Again :  suppose  it  be  required,  in  a  frame  building,  to 
find  the  length  of  a  brace  having  a  run  of  three  feet  each 
way  from  the  point  of  the  right  angle.  The  length  of  the 
sides  containing  the  right  angle  will  be  each  3  feet ;  then,  as 
before — 

3 
_3 

9  =  square  of  one  side. 
3  times  3  =  9  =  square  of  the  other  side. 

1 8  Product :  the  square  root  of  which  is  4  •  2426-}-  ft., 
or  4  feet  2  inches  and  -J  full. 

(2.) — The  hypothenuse  and  one  side  being  known,  to  find 
the  other  side. 

Rule. — Subtract  the  square  of  the  given  side  from  the 
square  of  the  hypothenuse,  and  the  square  root  of  the  prod- 
uct will  be  the  answer. 

Suppose  it  were  required  to  ascertain  the  greatest  per- 
pendicular height  a  roof  of  a  given  span  may  have,  when 
pieces  of  timber  of  a  given  length  are  to  be  used  as  rafters. 
Let  the  span  be  20  feet,  and  the  rafters  of  3  x  4  hemlock 
joist.  These  come  about  13  feet  long.  The  known  hy- 
pothenuse, then,  is  13  feet,  and  the  known  side,  10  feet — 
that  being  half  the  span  of  the  building. 

13 
13 

39 
13 

169  =  square  of  hypothenuse. 
10  times  10  =  100  —  square  of  the  given  side. 

69  Product :    the    square    root   of   which    is 


580  PRACTICAL   GEOMETRY. 

8  •  3066+  feet,  or  8  feet  3  inches  and  |  full.  This  will  be  the 
greatest  perpendicular  height,  as  required.  Again  :  suppose 
that  in  a  story  of  8  feet,  from  floor  to  floor,  a  step-ladder  is 
required,  the  strings  of  which  are  to  be  of  plank  12  feet 
long,  and  it  is  desirable  to  know  the  greatest  run  such  a 
length  of  string  will  afford.  In  this  case,  the  two  given 
sides  are — hypothenuse  12,  perpendicular  8  feet. 

12  times  12  —  144  —  square  of  hypothenuse, 
8  times    8  =    64  =  square  of  perpendicular. 

80  Product :  the  square  root  of  which  is 
8-9442+  feet,  or  8  feet  n  inches  and  ^ — the  answer,  as  re- 
quired. 

Many  other  cases  might  be  adduced  to  show  the  utility 
of  this  problem.  A  practical  and  ready  method  of  ascer- 
taining the  length  of  braces,  rafters,  etc.,  when  not  of  a  great 
length,  is  to  apply  a  rule  across  the  carpenters' -square. 
Suppose,  for  the  length  of  a  rafter,  the  base  be  12  feet  and 
the  height  7.  Apply  the  rule  diagonally  on  the  square,  so 
that  it  touches  12  inches  from  the  corner  on  one  side,  and  7 
inches  from  the  corner  on  the  other.  The  number  of  inches 
on  the  rule  which  are  intercepted  by  the  sides  of  the  square, 
13!-,  nearly,  wilt  be  the  length  of  the  rafter  in  feet  ;  viz.,  13 
feet  and. -J-  of  a  foot.  If  the  dimensions  are  large,  as  30  feet 
and  20,  take  the  half  of  each  on  the  sides  of  the  square,  viz., 
15  and  10  inches;  then  the  length  in  inches  across  will  be 
one  half  the  number  of  feet  the  rafter  is  long.  This  method 
is  just  as  accurate  as  the  preceding  ;  but  when  the  length  of 
a  very  long  rafter  is  sought,  it  requires  great  care  and  pre- 
cision to  ascertain  the  fractions.  For  the  least  variation  on 
the  square,  or  in  the  length  taken  on  the  rule,  would  make 
perhaps  several  inches  difference  in  the  length  of  the  rafter. 
For  shorter  dimensions,  however,  the  result  will  be  true 
enough. 

537. — To  Make  a  Circle  Equal  to  two  Given  Circles.— 

Let  A  and  B  (Fig-  396)  be  the  given  circles.  In  the  right- 
angled  triangle  abc  make  ab  equal  to  the  diameter  of  the 


SIMILAR  FIGURES. 


circle  B,  and  cb  equal  to  the  diameter  of  the  cin 

the  hypothenuse  a  c  will  be  the  diameter  of  a  circle  C,  which 

will  be  equal  in  area  to  the  two  circles  A  and  B,  added 

together. 


FIG.  396. 

Any  polygonal  figure,  as  A  (Fig.  397),  formed  on  the  hy- 
pothenuse of  a  right-angled  triangle,  will  be  equal  to  two 
similar  figures,*  as  B  and  C,  formed  on  the  two  legs  of  the 
triangle. 


FIG.  397. 

538. — To  €on§truct  a  Square  Equal  to  a  Given  Rect- 
angle.— Let  A  (Fig.  398)  be  the  given  rectangle.  Extend 
the  side  ab  and  make  be  equal  to  be\  bisect  a  c  in  /,  and 
upon  /,  with  the  radius  fa,  describe  the  semi-circle  agc\ 
extend  eb  till  it  cuts  the  curve  in  g\  then  a  square  bghd, 
formed  on  the  line  bg,  will  be  equal  in  area  to  the  rectan- 
glcA. 

*  Similar  figures  are  such  as  have  their  several  angles  respectively  equal, 
and  their  sides  respectively  proportionate. 


582 


PRACTICAL   GEOMETRY. 


Another  method.  Let  A  (Fig.  399)  be  the  given  rectangle. 
Extend  the  side  a  b  and  make  a  d  equal  to  a  c  ;  bisect  a  d  in 
e\  upon  e,  with  the  radius  ea,  describe  the  semi-circle  afd\ 
extend  gb  till  it  cuts  the  curve  in  /;  join  a  and  f\  then 


FIG.  398. 

the  square  B,  formed  on  the  line  af,  will  be  equal  in  area  to 
the  rectangle  A.     (See  Arts.  352  and  353.) 

539. — To  Form  a  Square  Equal  to  a  Given  Triangle- 
Let  ab  (Fig.  398)  equal  the  base  of  the  given  triangle,  and  be 


equal  half  its  perpendicular  height  (see  Fig.  392) ;  then  pro- 
ceed as  directed  at  Art.  538. 


540.— Two  Right  Lines  being  Given,  to  Find  a  Third 
Proportional  Thereto. — Let  A  and  B  (Fig.  400)  be  the  given 
lines.  Make  a  b  equal  to  A  ;  from  a  draw  a  c  at  any  angle 


PROPORTIONATE  DIVISIONS   IN   LINES.  583 

with  ab-,  make  ac  and  ad  each  equal  to  B;  join  c  and  £; 
from  d  draw  de  parallel  to  c  b ;  then  #  e  will  be  the  third 
proportional  required.  That  is,  ae  bears  the  same  propor- 
tion to  B  as  B  does  to  A. 


FIG.  400. 


541. — Three  Right  Lines  being  Given,  to  Find  a  Fourth 
Proportional  Thereto.— Let  A,  B,  and  C  (Fig.  401)  be  the 
given  lines.  Make  ab  equal  to  A  ;  from  a  draw  ac  at  any 


angle  with  a  b  ;  make  #  c  equal  to  j9  and  a  e  equal  to  (7 ;  join 
c  and  £;  from  e  draw  */  parallel  to  cb\  then  0/  will  be  the 
fourth  proportional  required.  That  is,  af  bears  the  same 
proportion  to  C  as  B  does  to  A. 

To  apply  this  problem,  suppose  the  two  axes  of  a  given 
ellipsis  and  the  longer  axis  of  a  proposed  ellipsis  are  given. 
Then,  by  this  problem,  the  length  of  the  shorter  axis  to  the 
proposed  ellipsis  can  be  found  ;  so  that  it  will  bear  the  same 
proportion  to  the  longer  axis  as  the  shorter  of  the  given 
ellipsis  does  to  its  longer.  (See  also  Art.  559.) 

542. — A  Line  with  Certain  Divisions  being  Given,  to 
Divide  Another,  Longer  or  Shorter,  Given  Line  in  the 
Same  Proportion. — Let  A  (Fig.  402)  be  the  line  to  be  di- 
vided, and  B  the  line  with  its  divisions.  Make  a  b  equal  to 
B  with  all  its  divisions,  as  at  i,  2,  3',  etc.;  from  a  draw  ac  at 
any  angle  with  a  b ;  make  a  c  equal  to  A  ;  join  c  and  b  ;  from 


584 


PRACTICAL   GEOMETRY. 


the  points  i,  2,  3,  etc.,  draw  lines  parallel  to  cb',  then  these 
will  divide  the  line  ac  in  the  same  proportion  as  B  is  divided 
— as  was  required. 

This  problem  will  be  found  useful  in  proportioning  the 


members  of  a  proposed  cornice,  in  the  same  proportion  as 
those  of  a  given  cornice  of  another  size.  (See  Art.  321.)  So 
of  a  pilaster,  architrave,  etc. 

543. — Between  Two  Given  Right  Lines,  to  Find  a 
Mean  Proportional. — Let  A  and  B  (Fig.  403)  be  the  given 
lines.  On  the  line  ac  make  ab  equal  to  A  and  be  equal  to 
B  ;  bisect  ac  in  e\  upon  e,  with  ea  for  radius,  describe  the 
semi-circle  adc\  at  b  erect  b d  at  right  angles  to  a c ;  then 


bd  will  be  the  mean  proportional  between  A  and  B.  That 
is,  ab  is  to  bd  as  bd  is  to  be.  This  is  usually  stated  thus: 
ab  :  bd  :  :  bd  :  be,  and  since  the  product  of  the  means 
equals  the  product  of  the  extremes,  therefore,  abxbe  =  bd*- 
This  is  shown  geometrically  at  Art.  538. 


CONIC    SECTIONS. 


544. — Definitions.  —  If  a   cone,   standing  upon  a  base 
that  is  at  right  angles  with  its  axis,  be  cut  by  a  plane,  per- 


AXIS  AND   BASE  OF  PARABOLA. 


585 


pendicular  to  its  base  and  passing  through  its  axis,  the  sec- 
tion will  be  an  isosceles  triangle  (as  a  be,  Fig.  404) ;  and  the 
base  will  be  a  semi-circle.  If  a  cone  be  cut  by  a  plane  in  the 
direction  ef  the  section  will  be  an  ellipsis ;  if  in  the  direction 
ml,  the  section  will  be  a  parabola;  and  if  in  the  direction 
ro,  an  hyperbola.  (See  Art.  499.)  If  the  cutting  planes  be 
at  right  angles  with  the  plane  a  be,  then— 

545. — To  Find  the  Axe§  of  the  Ellipsi§:   bisect  ef  (Fig. 
404)  in  g\  through  g  draw  h  i  parallel  to  ab\  bisect  h  i  in/; 


FIG.  404. 

upon  j,  with  jh  for  radius,  describe  the  semi-circle  hki\ 
from  ^-draw  gk.at  right  angles  to  hi\  then  twice  gk  will 
be  the  conjugate  axis  and  ef  the  transverse. 

546. — To  Find  the  Axis   and  Ba§e  of  the  Parabola. — 

Let  ;;/  /  (Fig.  404),  parallel  to  ac,  be  the  direction  of  the  cut- 
ting plane.  From  ;;/  draw  m  d  at  right  angles  to  a  b ;  then 
l-m  will  be  the  axis  and  height,  and  md  an  ordinate  and  half 
the  base,  as  at  Figs.  417,  418. 

547. — To  Find  the  Height,  Ba§e,  and  Transverse  Axis 
of  an  Hyperbola. — Let  o  r  (Fig.  404)  be  the  direction  of  the 


586 


PRACTICAL   GEOMETRY. 


cutting  plane.  Extend  or  and  ac  till  they  meet  at  w;  from 
o  draw  op  at  right  angles  to  a  b ;  then  r  o  will  be  the  height, 
n  r  the  transverse  axis,  and  op  half  the  base  ;  as  at  Fig.  419. 

54-8. — The  Axes  being  Given,  to  Find  the  Foci,  and  to 
Describe  an  Ellipsis  with  a  String. — Let  ab  (Fig.  405)  and 
cd  be  the  given  axes.  Upon  c,  with  a  e  or  be  for  radius,  de- 
scribe the  arc  //;  then  /  and  /,  the  points  at  which  the 
arc  cuts  the  transverse  axis,  will  be  the  foci.  At  f  and  f 
place  two  pins,  and  another  at  c\  tie  a  string  about  the  three 
pins,  so  as  to  form  the  triangle  ffc ;  remove  the  pin  from  c 
and  place  a  pencil  in  its  stead ;  keeping  the  string  taut, 


move  the  pencil  in  the  direction  cga\  it  will  then  describe 
the  required  ellipsis.  The  lines  fg  and  gf  show  the  posi- 
tion of  the  string  when  the  pencil  arrives  at  g. 

This  method,  when  performed  correctly,  is  perfectly  ac- 
curate ;  but  the  string  is  liable  to  stretch,  and  is,  therefore, 
not  so  good  to  use  as  the  trammel.  In  making  an  ellipse  by 
a  string  or  twine,  that  kind  should  be  used  which  has  the 
least  tendency  to  elasticity.  For  this  reason,  a  cotton  cord, 
such  as  chalk-lines  are  commonly  made  of,  is  not  proper  for 
the  purpose ;  a  linen  or  flaxen  cord  is  much  better. 


549. — The  Axes  being  Given,  to  Describe  an  Ellipsis 
with  a  Trammel. — Let  ab  and  cd  (Fig.  406)  be  the  given 
axes.  Place  the  trammel  so  that  a  line  passing  through  the 
centre  ol  the  grooves  would  coincide  with  the  axes  ;  make 


ELLIPSE   BY   TRAMMEL. 


587 


the  distance  from  the  pencil  e  to  the  nut/  equal  to  half  c d\ 
also,  from  the  pencil  e  to  the  nut  g  equal  to  half  a  b ;  letting 
the  pins  under  the  nuts  slide  in  the  grooves,  move  the  tram- 
mel eg  in  the  direction  cbd\  then  the  pencil  at  e  will  de- 
scribe the  required  ellipse. 

A  trammel  may  be  constructed  thus :  take  two  straight 
strips  of  board,  and  make  a  groove  on  their  face,  in  the  cen- 
tre of  their  width  ;  join  them  together,  in  the  middle  of  their 
length,  at  right  angles  to  one  another ;  as  is  seen  at  Fig.  406. 
A  rod  is  then  to  be  prepared,  having  two  movable  nuts 
made  of  wood,  with  a  mortise  through  them  of  the  size  of 
the  rod,  and  pins  under  them  large  enough  to  fill  the 
grooves.  Make  a  hole  at  one  end  of  the  rod,  in  which  to 


FIG.  406. 


place  a  pencil.  In  the  absence  of  a  regular  trammel  a  tem- 
porary one  may  be  made,  which,  for  any  short  job,  will  an- 
swer every  purpose.  Fasten  two  straight-edges  at  right 
angles  to  one  another.  Lay  them  so  as  to  coincide  with  the 
axes  of  the  proposed  ellipse,  having  the  angular  point  at  the 
centre.  Then,  in  a  rod  having  a  hole  for  the  pencil  at  one 
end,  place  two  brad-awls  at  the  distances  described  at  Art, 
549.  While  the  pencil  is  moved  in  the  direction  of  the 
curve,  keep  the  brad-awls  hard  against  the  straight-edges, 
as  directed  for  using  the  trammel-rod,  and  one  quarter  of 
the  ellipse  will  be  drawn.  Then,  by  shifting  the  straight- 
edges, the  other  three  quarters  in  succession  may  be  drawn. 
If  the  required  ellipse  be  not  too  large,  a  carpenters'-square 
may  be  made  use  of,  in  place  of  the  straight-edges. 

An  improved  method  of  constructing  the  trammel  is  as 


588 


PRACTICAL    GEOMETRY. 


follows:  make  the  sides  of  the  grooves  bevelling  from  the 
face  of  the  stuff,  or  dove-tailing  instead  of  square.  Prepare 
two  slips  of  wood,  each  about  two  inches  long,  which  shall 
be  of  a  shape  to  just  fill  the  groove  when  slipped  in  at  the 
end.  These,  instead  of  pins,  are  to  be  attached  one  to  each 
of  the  movable  nuts  with  a  screw,  loose  enough  for  the  nut 
to  move  freely  about  the  screw  as  an  axis.  The  advantage 
of  this  contrivance  is,  in  preventing  the  nuts  from  slipping 
out  oftheir  places  during  the  operation  of  describing  the 
curve. 

550. — To   Describe  an   Ellipsis  by   Ordiiiute*. — Let  ab 

and  cd  (Fig.  407)  be  given  axes.      With  c  e  or '  e  d  for  radius 


describe  the  quadrant  fgh  ;  divide  f/i,  ac,  and  eb,  each  into 
a  like  number  of  equal  parts,  as  at  I,  2,  and  3  ;  through 
these  points  draw  ordinates  parallel  to  cd  and  -fg\  take  the 
distance  I  i  and  place  it  at  i  /,  transfer  27  to  2  /#,  and  3  k  to 
3  n ;  through  the  points  # ,  n,  m,  /,  and  c,  trace  a  curve,  and 
the  ellipsis  will  be  completed. 

The  greater  the  number  of  divisions  on  a,  e,  etc.,  in  this 
and  the  following  problem,  the  more  points  in  the  curve  can 
be  found,  and  the  more  accurate  the  curve  can  be  traced. 
If  pins  are  placed  in  the  points  n,  m,  /,  etc.,  and  a  thin  slip 
of  wood  bent  around  by  them,  the  curve  can  be  made  quite 
correct.  This  method  is  mostly  used  in  tracing  face-moulds 
for  stair  hand -railing. 

551. — To  Describe  an  Ellipsis  by  Intersection  of  Lines. 

— Let  ab  and  cd  (Fig.  408)  be  given  axes.     Through  c,  draw 


ELLIPSE  BY  INTERSECTION  OF  LINES. 


589 


fg  parallel  to.ab',  from  a  and  b  draw  af  and  b  g  at  right 
angles  to  a  b ;  divide  fa,  gb,  ae,  and  eb,  each  into  a  like 
number  of  equal  parts,  as  at  i,  2,  3,  and  0,  <?,<?;  from  I,  2, 
and  3,  draw  lines  to  c;  through  0,  0,  and  0,  draw  lines  from  d, 


intersecting  those  drawn  to  c ;  then  a  curve,  traced  through 
the  points  i,  i,  i,  will  be  that  of  an  ellipsis. 

Where  neither  trammel  nor  string  is  at  hand,  this,  per- 
haps, is  the  most  ready  method  of  drawing  an  ellipsis.  The 
divisions  should  be  small,  where  accuracy  is  desirable.  By 
this  method  an  ellipsis  may  be  traced  without  the  axes,  pro- 
vided that  a  diameter  and  its  conjugate  be  given.  Thus,  ab 


and  cd  (Fig.  409)  are  conjugate  diameters:  fg\s  drawn  par- 
allel to  ab,  instead  of  being  at  right  angles  to  cd\  also,  fa 
and  g  b  are  drawn  parallel  to  c  d,  instead  of  being  at  right 
angles  to  a  b. 


590 


PRACTICAL   GEOMETRY. 


552. — To  Describe  an  Ellipsis  by  Intersecting  Arcs. — 

Let  a  b  and  cd  (Fig.  410)  be  given  axes.  Between  one  of  the 
foci,  f  and  f,  and  the  centre  e,  mark  any  number  of  points, 
at  random,  as  I,  2,  and  3  ;  upon  f  and  /,  with  b  i  for  radius, 
describe  arcs  at  gt  g,  g,  and  g ;  upon  f  and  /,  with  a  I  for 


radius,  describe  arcs  intersecting  the  others  at  g>g,g,  and  g; 
then  these  points  of  intersection  will  be  in  the  curve  of  the 
ellipsis.  The  other  points,  h  and  i,  are  found  in  like  manner, 
viz.:  h  is  found  by  taking  b2  for  one  radius,  and  0,2  for  the 
other ;  i  is  found  by  taking  b  3  for  one  radius,  and  a  3  for  the 


other,  always  using  the  foci  for  centres.  Then  by  tracing  a 
curve  through  the  points  c,  g,  //,  i,  b,  etc.,  the  ellipse  will  be 
completed. 

This  problem  is  founded  upon  the  same  principle  as  that 
of  the  string.  This  is  obvious,  when  we  reflect  that  the 
length  of  the  string  is  equal  to  the  transverse  axis,  added  to 


TO   DESCRIBE   AN   OVAL. 


591 


the  distance  between  the  foci.     See  Fig.  405,  in  which  cf 
equals  ae,  the  half  of  the  transverse  axis. 

553. — To  Describe  a  Figure  Nearly  in  the  Shape  of  an 
Ellipsis,  by  a  Pair  of  Compasses. — Let  ab  and  c  d  (Fig.  41 1) 
be  given  axes.  From  c  draw  c  e  parallel  to  a  b  ;  from  a  draw 
ae  parallel  to  cd\  join  e  and  d\  bisect  ea  in  /;  join  /  and  c, 
intersecting  edvn.  i\  bisect  ic  in,*?;  from  o  draw  og  at  right, 
angles  to  ic,  meeting  cd  extended  to  g\  join  i  and  g,  cutting 
the  transverse  axis  in  r ;  make  hj  equal  to  Jig,  and  h  k  equal 
to//r;  from  j,  through  r  and  k,  draw//;z  and/#;  also,  from 
g,  through  /£,  draw  gl;  upon  g  and/,  with  gc  for  radius, 
describe  the  arcs  il  and  mn\  upon  r  and  £,  with  ?-#  for 


radius,  describe  the  arcs  ;;/*and  /;/ ;  this  will  complete  the 
figure. 

When  the  axes  are  proportioned  to  one  another,  as  at  2 
to  3,  the  extremities,  c  and  d,  of  the  shortest  axis,  will  be 
the  centres  for  describing  the  arcs  il  and  m  n  ;  and  the  inter- 
section of  ed  with  the  transverse  axis  will  be  the  centre  for 
describing  the  arc  m,  i,  etc.  As  the  elliptic  curve  is  contin- 
ually changing  its  course  from  that  of  a  circle,  a  true  ellipsis 
cannot  be  described  with  a  pair  of  compasses.  The  above, 
therefore,  is  only  an  approximation. 

554. — To  Draw  an  Oval  in  the  Proportion  Seven  by 
Nine. — Let  cd  (Fig.  412)  be  the  given  conjugate  axis.  Bisect 


592 


PRACTICAL  GEOMETRY. 


cdin  o,  and  through  o  draw  ab  at  right  angles  to  cd\  bisect 
co  in  e ;  upon  o,  with  0^  for  radius,  describe  the  circle  efgh  ; 
from  e,  through  h  and  ft  draw  <?/  and  ei\  also,  from  ^, 
through  h  and  /,  draw  ^£  and  gl\  upon  £•,  with  gc  for 
radius,  describe  the  arc  kl]  upon  *,  with  e d  for  radius,  de- 
scribe the  arc  ji ;  upon  h  and  /,  with  h  k  for  radius,  describe 
the  arcs  jk  and  /*';  this  will  complete  the  figure. 


This  is  an  approximation  to  an  ellipsis ;  and  perhaps  no 
method  can  be  found  by  which* a  well-shaped  oval  can  be 
drawn  with  greater  facility.  By  a  little  variation  in  the 
process,  ovals  of  different  proportions  may  be  obtained.  If 
quarter  of  the  transverse  axis  is  taken  for  the  radius  of  the 
circle  efgh,  one  will  be  drawn  in  the  proportion  five  by 
seven. 


FIG.  414. 

555. — To  Draw  a  Tangent  to  an  Ellipsis. — Let  abed 
(Fig.  413)  be  the  given  ellipsis,  and  d  the  point  of  contact. 
Find  the  foci  (Art.  548)  /  and  ft  and  from  them,  through  d, 
draw  fe  and  fd\  bisect  the  angle  (Art.  506)  edo  with  the 
line  sr ;  then  sr  will  be  the  tangent  required. 


TO    FIND   THE  AXES   OF  AN   ELLIPSE. 


593 


556. — An  Ellipsis  with  a  Tangent  Given,  to  Detect  the 
Point  of  Contact. — Let  a gb  f  (Fig.  414)  be  the  given  ellip- 
sis and  tangent.  Through  the  centre  e  draw  a  b  parallel  to 
the  tangent;  anywhere  between  e  and  /  draw  cd  parallel  to 
a  b ;  bisect  cd  in  o ;  through  o  and  e  draw  fg]  then  g  will 
be  the  point  of  contact  required. 

557. — A  Diameter  of  an  Ellipsi§  Given,  to  Find  its 
Conjugate. — Let  a  b  (Fig.  414)  be  the  given  diameter.  Find 
the  line  fg  by  the  last  problem ;  then  fg  will  be  the  diam- 
eter required. 

558. — Any  Diameter  and  its  Conjugate  being  Given,  to 
Ascertain  the  Two  Axes,  and  thence  to  Describe  the  Ellipsis. 

—Let  a  b  and  cd(Fig.  415)  be  the  given  diameters,  conjugate 


FIG.  415. 

to  one  another.  Through  c  draw  cf  parallel  to  a  b ;  from  c 
draw  eg  at  right  angles  to  ef\  make  eg  equal  to  ah  or  Jib\ 
join  g  and  h ;  upon  g,  with  gc  for  radius,  describe  the  arc 
ikcj\  upon  h,  with  the  same  radius,  describe  the  arc  In  • 
through  the  intersections  /  and  n  draw  n  o,  cutting  the  tan- 
gent ef  in  o  ;  upon  o,  with  ogfor  radius,  describe  the  semi- 
circle  e  igf\  join  e  and  g,  also  g  and  f,  cutting  the  arc  icj 
in  k  and  / ;  from  r,  through  //,  draw  e  m,  also  from  /,  through 
h,  draw  // ;  from  k  and  t  draw  kr  and  ts  parallel  to  gk 


594 


PRACTICAL   GEOMETRY. 


cutting  em  in  r,  and  //  in  s ;  make  h m  equal  to  hr,  and  hp 
equal  to  hs\  then  r;;/  and  s p  will  be  the  axes  required,  by 
which  the  ellipsis  may  be  drawn  in  the  usual  way. 

559. — To  Describe  an  Ellipsis,  whose  Axes  shall  toe 
Proportionate  to  the  Axes  of  a  Larger  or  SmalDer  Given 
One. — Let  a cbd(Fig.  416)  be  the  given  ellipsis  and  axes,  and 


FIG.  416. 

ij  the  transverse  axis  of  a  proposed  smaller  one.  Join  a  and 
c\  from  i  draw  ie  parallel  to  ac ;  make  of  equal  to  oe ;  then 
ef  will  be  the  conjugate  axis  required,  and  will  bear  the 
same  proportion  to  ij  as  cd does  to  ab.  (See  Art.  541.) 

560. — To  Describe  a  Parabola  by  Intersection  of  Lines. 

— Let  ml  (Fig.  417)  be  the  axis  and  height  (see  Fig.  404)  and 


i    2 


/      3     2      1 


1      2     3    in    3    2 

FIG.  417. 


dd  a  double  ordinate  and  base  of  the  proposed  parabola. 
Through  /  draw  a  a  parallel  to  dd\  through  d  and  d  draw 
da  and  da  parallel  to  ml\  divide  ad  and  dm,  each  into  a 
like  number  of  equal  parts  ;  from  each  point  of  division  in 


TO   DESCRIBE   AN    HYPERBOLA. 


595 


dm  draw  the  lines  i  i,  22,  etc.,  parallel  to  ;;//;  from  each 
point  of  division  in  da  draw  lines  to/;  then  a  curve  traced 
through  the  points  of  intersection  o,  o,  and  o,  will  be  that  of 
a  parabola. 

Another  method.  Let  m  I  (Fig.  418)  be  the  axis  and  height, 
and  dd  the  base.  Extend  m  I  and  make  la  equal  to  m  I ; 
join  a  and  d,  and  a  and  d\  divide  ad  and  ad,  each  into  a 
like  number  of  equal  parts,  as  at  i,  2,  3,  etc. ;  join  i  and  i,  2 
and  2,  etc.,  and  the  parabola  will  be  completed.  (See  Arts. 
460  to  472.) 

561. — To  Describe  an  Hyperbola  by  Intergection  of 
Lines. — Let  ro  (Fig.  419)  be  the  height,//  the  base,  and  nr 
the  transverse  axis.  (See  Fig.  404.)  Through  r  draw  a  a 


m 

FIG.  418. 


1    2    3   o    3    2     \    p 

FIG.  419. 


parallel  to  pp\  from  /  draw  ap  parallel  to  ro\  divide  ap 
and  po,  each  into  a  like  number  of  equal  parts  ;  from  each 
of  the  points  of  division  in  the  base,  draw  lines  to  ;/ ;  from 
each  of  the  points  of  division  in  ap,  draw  lines  to  r\  then 
a  curve  traced  through  the  points  of  intersection  o,  o,  etc., 
will  be  that  of  an  hyperbola. 

The  parabola  and  hyperbola  afford  handsome  curves  for 
various  mouldings.  (See  Figs.  191  to  205  ;  222  to  224;  241 
and  242  ;  also  note  to  Art.  318.) 


SECTION   XVII.— SHADOWS. 

562. — The  Art  of  Drawing  consists  in  representing 
solids  Upon  a  plane  surface,  so  that  a  curious  and  nice  ad- 
justment of  lines  is  made  to  present  the  same  appearance  to 
the  eye  as  does  the  human  figure,  a  tree,  or  a  house.  It  is 
by  the  effects  of  light,  in  its  reflection,  shade,  and  shadow, 
that  the  presence  of  an  object  is  made  known  to  us;  so 
upon  paper  it  is  necessary,  in  order  that  the  delineation 
may  appear  real,  to  represent  fully  all  the  shades  and  shad- 
ows that  would  be  seen  upon  the  object  itself.  In  this  sec- 
tion I  propose  to  illustrate,  by  a  few  plain  examples,  the 
simple  elementary  principles  upon  which  shading,  in  archi- 
tectural subjects,  is  based.  The  necessary  knowledge  of 
drawing,  preliminary  to  this  subject,  is  treated  of  in  Section 
XV.,  from  Arts.  487  to  498. 

563. — The  Inclination  of  the  JLlne  of  Shadow. — This 
is  always,  in  architectural  drawing,  45  degrees,  both  on  the 
elevation  and  on  the  plan ;  and  the  sun  is  supposed  to  be 
behind  the  spectator,  and  over  his  left  shoulder.  This  can 
be  illustrated  by  reference  to  Fig.  420,  in  which  A  repre- 
sents a  horizontal  plane,  and  B  and  C  two  vertical  planes 
placed  at  right  angles  to  each  other.  A  represents  the  plan, 
C  the  elevation,  and  B  a  vertical  projection  from  the  eleva- 
tion. In  finding  the  shadow  of  the  plane  B,  the  line  a  b  is 
drawn  at  an  angle  of  45  degrees  with  the  horizon,  and  the 
liner^  at  the  same  angle  with  the  vertical  planed.  The 
plane  B  being  a  rectangle,  this  makes  the  true  direction  of 
the  sun's  rays  to  be  in  a  course  parallel  to  db,  which  direc- 
tion has  been  proved  to  be  at  an  angle  of  35  degrees  and 
16  minutes  with  the  horizon.  It  is  convenient,  in  shading, 
to  have  a  set-square  with  the  two  sides  that  contain  the 


CONVENTIONAL   PLANES    OF   SHADOW. 


597 


right  angle  of  equal  length;  this  will  make  the  two  acute 
angles  each  45  degrees,  and  will  give  the  requisite  bevel 
when  worked  upon  the  edge  of  the  T-square.  One  reason 
why  this  angle  is  chosen  in  preference  to  another  is  that 
when  shadows  are  properly  made  upon  the  drawing  by  it, 
the  depth  of  every  recess  is  more  readily  known,  since  the 
breadth  of  shadow  and  the  depth  of  the  recess  will  be  equal. 
To  distinguish  between  the  terms  shade  and  shadow,  it  will 
be  understood  that  all  such  parts  of  a  body  as  are  not  e5c- 
posed  to  the  direct  action  of  the  sun's  rays  are  in  shade ; 


while  those  parts  which  are  deprived  of  light  by  the  inter- 
position of  other  bodies  are  in  shadow. 

564. — To  Find  the  Line  of  Shadow  011  Mouldings  and 
other  Horizontally  Straight  Projections. — Figs.  421,  422, 
423,  and  424  represent  various  mouldings  in  elevation,  re- 
turned at  the  left,  in  the  usual  manner  of  mitering  around  a 
projection.  A  mere  inspection  of  the  figures  is  sufficient  to 
see  how  the  line  of  shadow  is  obtained,  bearing  in  mind  that 
the  ray  a  b  is  drawn  from  the  projections  at  an  angle  of  45 


598 


SHADOWS. 


degrees.  When  there  is  no  return  at  the  end,  it  is  neces- 
sary to  draw  a  section,  at  any  place  in  the  length  of  the 
mouldings,  and  find  the  line  of  shadow  from  that. 

565. — To  Find  tlie  Line  of  Shadow  Cast  by  a  Shelf. — In 

Fig.  425,  A  is  the  plan  and  B  is  the  elevation  of  a  shelf 
attached  to  a  wall.  From  a  and  c  draw  a  b  and  c  d,  accord- 
ing to  the  angle  previously  directed  ;  from  b  erect  a  per- 
pendicular intersecting  c  d  <&  d\  from  d  draw  de  parallel  to 


FIG.  421, 


FIG.  422. 


FIG.  423. 


FIG.  424. 


the  shelf;  then  the  lines  cd  and  de  will  define  the  shadow 
cast  by  the  shelf.  There  is  another  method  of  finding  the 
shadow,  without  the  plan  A.  Extend  the  lower  line  of  the 
shelf  to //and  make  cf  equal  to  the  projection  of  the  shelf 
from  the  wall  ;  from/ draw  fg  at  the  customary  angle,  and 
from  c  drop  the  vertical  line  eg  intersecting  fg^g\  from 
g  draw  ge  parallel  to  the  shelf,  and  from  c  draw  c  d  at  the 
usual  angle;  then  the  lines  cd  and  de  will  determine  the 
extent  of  the  shadow  as  before. 


SHADOWS   OF   STRAIGHT  AND   OBLIQUE   SHELVES.       599 

.566.— To  Find  the  Shadow  Cast  by  a  Shelf  which  i* 
Wider  at  one  End  than  at  the  Other. — In  Fig.  426,  A  is  the 
plan,  and  B  the  elevation.  Find  the  point  d,  as  in  the  pre- 


B 


FIG.  425. 


vious  example,  and  from  any  other  point  in  the  front  of  the 
shelf,  as  a,  erect  the  perpendicular  a  e ;  from  a  and  e  draw  a  b 
and  e  c,  at  the  proper  angle,  and  from  b  erect  the  perpendicu- 


FIG.  426. 


lar  be,  intersecting  ec  in  c\  from  d,  through  c,  dra\v  do\ 
then  the  lines  id  and  do  will  give  the  limit  of  the  shadow 
cast  by  the  shelf. 


6oo 


SHADOWS. 


567.— To  Find  tlie  Shadow  of  a  Shelf  having  one  End 
Aeute  or  Obtuse  Angled. — Fig.  427  shows  the  plan  and  ele- 
vation of  an  acute-angled  shelf.  Find  the  line  eg  as  before  ; 


FIG.  427. 

from  a  erect  the  perpendicular  ab\  join  b  and  e\  then  be 
and  eg  will  define  the  boundary  of  shadow. 

568.— To  Find  the  shadow  Cast  by  an  Inclined  Shelf.— 

In  Fig.  428  the  plan  and  elevation  of  such  a  shelf  are  shown, 
having  also  one  end  wider  than  the  other.     Proceed  as  di- 


FIG   428. 

rected  for  finding  the  shadows  of  Fig.  426,  and  find  the  points 
^/and  c ;  then  ad  and  dc  will  be  the  shadow  required.  If 
the  shelf  had  been  parallel  in  width  on  the  plan,  then  the 
line  dc  would  have  been  parallel  with  the  shelf  a  b. 


SHADOWS   OF   INCLINED   AND    CURVED   SHELVES.         6oi 

569.— To  Find  the  Shadow  Cast  by  a  Shelf  Inclined  in 
its  Vertical  Section  either  Upward  or  Downward. — From 
a  (Figs.  429  and  430)  draw  a  b  at  the  usual  angle,  and  from  b 
draw  be  parallel  with  the  shelf;  obtain  the  point  e  by  draw- 


FIG.  429. 


FIG.  430. 


ing -a  line  from  d  at  the  usual  angle.  In  Fig.  429  join  e  and 
i ;.  then  ic  and  cc  will  define  the  shadow.  In  Fig.  430,  from 
o  draw  oi  parallel  with  the  shelf  ;  join  i  and  e ;  then  ie  and 
cc  will  be  the  shadow  required. 

The  projections  in  these  several  examples  are  bounded 


FIG.  431- 


FIG.  432. 


by  straight  lines  ;  but  the  shadows  of  curved  lines  may  be 
found  in  the  same  manner,  by  projecting  shadows  from  sev- 
eral points  in  the  curved  line,  and  tracing  the  curve  .of 
shadow  through  these  points.  (Figs.  431  and  432.) 


602 


SHADOWS. 


570.— To  Find  the  Shallow  of  a  Shelf  having  its  Front 
Edge,  or  End,  Curved  on  the  Plan, — In  Figs.  431  and  432 
A  and  A  show  an  example  of  each  kind.  From  several 
points,  as  a,  a,  in  the  plan,  and  from  the  corresponding-  points 
0,  o  in  the  elevation,  draw  rays  and  perpendiculars  intersect- 


FIG.  433. 

ing  at  e,  <:,  etc. ;  through  these    points  of  intersection  trace 
the  curve,  and  it  will  define  the  shadow. 

57 L—  To  Find  the  Shadow  of  a  Shelf  Curved  in  the  Ele- 
vation.— In  Fig.  433  find  the  points  of  intersection,  e,  c  and 


FIG.  434 

e,  as  in  the  last  examples,  and  a  curve  traced  through  them 
will  define  the  shadow. 

The  preceding  examples  show  how  to  find  shadows  when 
cast  upon  a  vertical  plane  ;  shadows  thrown  upon  curved  sur- 
•  faces  are  ascertained  in  a  similar  manner.     (Fig.  434.) 


SHADOW   UPON  AN   INCLINED   WALL. 


603 


572.— To  Find  the  Shadow  Cast  upon  a  Cylindrical 
Wall  by  a  Projection  of  any  Kind.  —  By  an  inspection  of 
Fig.  434,  it  will  be  seen  that  the  only  difference  between  this 
and  the  last  examples  is  that  the  rays  in  the  plan  die  against 
the  circle  ab,  instead  of  a  straight  line. 


573.— To  Find  the  Shadow  Ca§t  by  a  Shelf  upon  an  In- 
clined Wall. — Cast  the  ray  ab  (Fig.  435)  from  the  end  of  the 
shelf  to  the  face  of  the  wall,  and  from  b  draw  be  parallel  to 
the  shelf;  cast  the  ray  de  from  the  end  of  the  shelf;  then 
the  lines  de  and  ec  will  define  the  shadow. 


FIG.  436. 

These  examples  might  be  multiplied,  but  enough  has 
been  given  to  illustrate  the  general  principle  by  which  shad- 
ows in  all  instances  are  found.  Let  us  attend  now  to  the 
application  of  this  principle  to  such  familiar  objects  as  are 
likely  to  occur  in  practice. 


004  SHADOWS. 

574.— To  Find  the  Shadow  of  a  Projecting  Horizontal 
Ream. — From  the  points  a,  a,  etc.  (Fig.  436),  cast  rays  upon 
the  wall ;  the  intersections  e,  e,  e  of  those  rays  with  the  per- 
pendiculars drawn  from  the  plan  will  define  the  shadow.  If 
the  beam  be  inclined,  either  on  the  plan  or  elevation,  at  any 
angle  other  than  a  right  angle,  the  difference  in  the  manner 


FIG.  437. 

of  proceeding  can  be  seen  by  reference  to  the  preceding 
examples  of  inclined  shelves,  etc. 

575. — To  Find  the  Shadow  in  a  Recess. — From  the  point 
a  (Fig.  437)  in  the  plan,  and  b  in  the  elevation,  draw  the  rays 
acand  be;  from  c  erect  the  perpendicular  ce,  and  from  e 


FIG.  438. 

draw  the  horizontal  line  ed;  then  the  lines  r^.and  ed  will 
show  the  extent  of  the  shadow.  This  applies  only  where 
the  back  of  the  recess  is  parallel  with  the  face  of  the  wall. 

576.— To  Find  the  Shadow  in  a  Rece§§,  when  the  Face 
of  the  Wall  Is  Inclined,  and  the  Baek  of  the  Recess  i§ 
Vertical. — In  Fig.  438,  A  shows  the  section  and  B  the  eleva- 


SHADOW   IN   A   FIREPLACE. 


605 


tion  of  a  recess  of  this  kind.  From  b,  and  from  any  other 
point  in  the  line  ba,  as  a,  draw  the  rays  be  and  ae ;  from  c, 
a,  and  e  draw  the  horizontal  lines  eg,  af,  and  eh;  from  d 


FIG.  439. 

and /cast  the  rays  di  and  ///;  from  i,  through  h,  draw  is; 
then  s  i  and  ig  will  define  the  shadow. 

577. TO  Find  the  Shadow  in  a  Fireplace. — From  a  and 

b  (Fig.  439)  cast  the  rays  a  c  and  b  e,  and  from  c  erect  the 


FIG.  440. 

perpendicular  cc\  from  c  draw  the  horizontal  line  eo,  and 
join  o  and  </;  then  c  c,  eo,  and  <?</  will  give  the  extent  of  the 
shadow. 


6o6 


SHADOWS. 


578.— To  Find  I  lie  Shadow  of  a  moulded  Window-Lin- 
tel.— Cast  rays  from  the  projections  a,  o,  etc.,  in  the  plan 
(Fig.  440),  and  d,  e,  etc.,  in  the  elevation,  and  draw  the  usual 
perpendiculars  intersecting  the  rays  at  z,  i,  and  i ;  these  in- 
tersections connected,  and  horizontal  lines  drawn  from  them, 
will  define  the  shadow.  The  shadow  on  the  face  of  the  lin- 
tel is  found  by  casting  a  ray  back  from  i  to  s,  and  drawing 
the  horizontal  line  s  n. 

579. — To  Find  the  Shadow  Cast  by  the  No§ing  of  a  Step. 

— From  a  (Fig.  441)  and  its  corresponding  point  c,  cast  the 


FIG.  441. 

rays  a  b  and  cd,  and  from  b  erect  the  perpendicular  b  d;  tan- 
gical  to  the  curve  at  e  cast  the  ray  ef,  and  from  c  drop  the 
perpendicular  e  o,  meeting  the  mitre-line  ag  in  o ;  cast  a  ray 
from  o  to  /',  and  from  /erect  the  perpendicular  if\  from  /t 
draw  the  ray  //  k  ;  from  /to  d  and  from  d  to  k  trace  the 
curve  as  shown  in  the  figure  ;  from  k  and  h  draw  the  hori- 
zontal lines  kn  and  hs\  then  the  limit  of  the  shadow  will  be 
completed. 

580.— To  Find  the  Shadow  Thrown  by  a  Pede§tal  upon 
Step§. — From  a  (Fig.  442)  in  the  plan,  and  from  c  in  the  ele- 
vation, draw  the  rays  ab  and  c  e  ;  then  ao  will  show  the  ex- 


SHADOWS   ON   STEPS   AND   COLUMNS. 


607 


tent  of  the  shadow  on  the  first  riser,  as  at  A  ;  fg  will  deter- 
mine the  shadow  on  the  second  riser,  as  at  B\  cd  gives  the 
amount  of  shadow  on  the  first  tread,  as  at  £7,  and  //  i  that  on 
the  second  tread,  as  at  D ;  which  completes  the  shadow  of 


FIG.  442. 


the  left-hand  pedestal,  both  on  the  plan  and  elevation.  A 
mere  inspection  of  the  figure  will  be  sufficient  to  show  how 
the  shadow  of  the  right-hand  pedestal  is  obtained. 


FIG.  443. 


FIG.  444. 


681.— To  Find  the  Shadow  Thrown  on  a  Column  by  a 
Square  Abacus. — From  a  and  b  (Fig.  443)  draw  the  rays  ac 
and  b  e,  and  from  c  erect  the  perpendicular  c  e ;  tangical  to 
the  curve  at  d  draw  the  ray  df,  and  from  //,  corresponding 
to /in  the  plan,  draw  the  ray  ho;  take  any  point  between  a 
and  fy  as  i,  and  from  this,  as  also  from  a  corresponding  point 


6o8 


SHADOWS. 


n,  draw  the  rays  ir  and  ns ;  from  r  and  from  d  erect  the 
perpendiculars  rs  and  do-,  through  the  points  e,  s,  and  o 
trace  the  curve  as  shown  in  the  figure  ;  then  the  extent  of 
the  shadow  will  be  defined. 


m 
X. 

N/^ 

A.    /" 

J 

^^^^^                                                ^TlWk 

FIG.  445. 

582. — To  Find  the  Shadow  Tin-own  on  a  Column  by  a 
Circular  Abacus. — This  is  so  nearly  like  the  last  example 
that  no  explanation  will  be  necessary,  farther  than  a  refer- 
ence to  the  preceding  article. 


SHADOWS   ON   THE   CAPITAL  OF  A   COLUMN". 


609 


583.— To  Find  the  Shadows  on  the  Capital  of  a  Columii. 

— This  may  be  done  according-  to  the  principles  explained 
in  the  examples  already  given  ;  a  quicker  way  of  doing  it, 
however,  is  as  follows :  if  we  take  into  consideration  one 
ray  of  light  in  connection  with  all  those  perpendicularly 
under  and  over  it,  it  is  evident  that  these  several  rays  would 
form  a  vertical  plane,  standing  at  an  angle  of  45  degrees 
with  the  face  of  the  elevation.  Now  we  may  suppose  the 
column  to  be  sliced,  so  to  speak,  with  planes  of  this  nature — • 
cutting  it  in  the  lines  a  b,  c  d,  etc.  (Fig.  445),  and,  in  the  ele- 


FIG.  446. 


vation,  find  by  squaring  up  from  the  plan,  the  lines  of  section 
which  these  planes  would  make  thereupon.  For  instance : 
in  finding  upon  the  elevation  the  line-of  section  a  I),  the  plane 
cuts  the  ovolo  at  e,  and  therefore  /  will  be  the  correspond- 
ing point  upon  the  elevation  ;  h  corresponds  with  g,  i  withy, 
o  with  s,  and  /  with  b.  Now,  to  find  the  shadows  upon  tfcis 
line  of  section,  cast  from  m  the  ray  m  «,  from  //  the  ray  h  o, 
etc. ;  then  that  part  of  the  section  indicated  by  the  letters 
m  f  i  n,  and  that  part  also  between  //  and  o  will  be  under 


6io 


SHADOWS. 


shadow.  By  an  inspection  of  the  figure,  it  will  be  seen  that 
the  same  process  is  applied  to  each  line  of  section,  and  in 
that  way  the  points  /,  r,  t,  u,  v,  ?v,  x,  as  also  i,  2,  3,  etc.,  are 


FIG.  447. 

successively  found,  and  the  lines  of  shadow  traced  through 
them. 

Fig.  446  is  an  example  of  the  same  capital  with  all  the 
shadows  finished  in  accordance  with  the  lines  obtained  on 
Fig.  445. 


SHADOW  OF  A  COLUMN  ON  A  WALL. 


584.— To  Find  the  Shadow  Thrown  on  a  Vertical  Wall 
l>y  a  Column  and  Entablature  Standing  in  Advance  of  §aid 
Wall. — Cast  rays  from  a  and  b  (Fig.  447),  and  find  the  point 
c  as  in  the  previous  examples  ;  from  d  draw  the  ray  de,  and 
from  e  the  horizontal  line  ef\  tangical  to  the  curve  at  g  and 
h  draw  the  rays  gj  and  h  i,  and  from  i  and  j  erect  the  per- 
pendiculars il  and/£;  from  m  and  n  draw  the  rays  mf  and 
nk,  and  trace  the  curve  between  £and/;  cast  a  ray  from  o  to 
/,  a  vertical  line  from/  to  s,  and  through  s  draw  the  horizon- 
tal line  s  t ;  the  shadow  as  required  will  then  be  completed. 


FIG.  448. 

Fig.  >| /| K  is  an  example  of  the  same  kind  as  the  last,  with 
all  the  shadows  filled  in,  according  to  the  lines  obtained  in 
the  preceding  figure. 

585. — Shadows  on  a  Cornice. — Figs.   449  and  450  are 

examples  of  the  Tuscan  cornice.     The  manner  of  obtaining 
the  shadows  is  evident. 

586. — Reflected  Light. — In  shading,  the  finish  and  life  of 
an  object  depend  much  on  reflected  light.  This  is  seen  to 
advantage  in  Fig.  446,  and  on  the  column  in  Fig.  448.  Re- 


6l2 


SHADOWS. 


fleeted  rays  are  thrown  in  a  direction  exactly  the  reverse 
of  direct  rays  ;  therefore,  on  that  part  of  an  object  which  is 
subject  to  reflected  light,  the  shadows  are  reversed.  The 


FIG.  449. 

fillet  of  the  ovolo  in  Fig.  446  is  an  example  of  this.  On  the 
right  hand  side  of  the  column,  the^ace  of  the  fillet  is  much 
darker  than  the  cove  directly  under  it.  The  reason  of  this 


FIG.  450, 


is,  the  face  of  the  fillet  is  deprived  both  of  direct  and  re- 
flected light,  whereas  the  cove  is  subject  to  the  latter.  Other 
instances  of  the  effect  of  reflected  light  will  be  seen  in  the 
other  examples. 


CONTENTS. 


PART  I. 


SECTION    I.  —  ARC  H  ITECTU  RE. 


Art.  1.  Building  defined,  p.  5.— 2.  Antique  Builcings  ;  Tower  of  Babel, 
p.  5. — 3.  Ancient  Cities  and  Monuments,  p.  6. — 4.  Architecture  in  Greece, 
p.  6. — 5.  Architecture  in  Rome,  p.  7. — 6.  Rome  and  Greece,  p.  8. — 7.  Ar- 
chitecture debased,  p.  9. — §.  The  Ostrogoths,  p.  9. — 9.  The  Lombards,  p.  10. 
— 1O.  The  Byzantine  Architects,  p.  10. — 11.  The  Moors,  p.  10. — 12.  The 
Architecture  of  England,  p.  n. — 13.  Architecture  Progressive,  p.  12. — 14. 
Architecture  in  Italy,  p.  12. — 15.  The  Renaissance,  p.  13. — 16.  Styles  of  Ar- 
chitecture, p.  13. — 17.  Orders,  p.  14.— 18.  The  Stylobatc,  p.  14. — 19.  The 
Column,  p.  14.— 2O.  The  Entablature,  p.  14.— 21.  The  Base,  p.  14—22. 
The  Shaft,  p.  15.— 23.  The  Capital,  p.  15.— 24.  The  Architrave,  p.  15.— 25. 
The  Frieze,  p.  15.— 26.  The  Cornice,  p.  15.— 27.  The  Pediment,  p.  15.— 28. 
The  Tympanum,  p.  15. — 29.  The  Attic,  p.  15. — 3O.  Proportions  in  an  Order, 
p.  15.— 31.  Grecian  Styles,  p.  16.— 32.  The  Doric  Order,  p.  16.— 33.  The 
Intercolumniation,  p.  17.— 34.  The  Doric  Order,  p.  19 — 35.  The  Ionic 
Order,  p.  19. — 36.  The  Intercolumniation,  p.  20. — 37.  To  Describe  the  Ionic 
Volute,  p.  20. — 38.  The  Corinthian  Order,  p.  23. — 39.  Persians  and  Carya- 
tides, p.  24. — 4O.  Persians,  p.  24. — 41,  Caryatides,  p.  26. — 42.  Roman 
Styles,  p.  26. — 43.  Grecian  Orders  modified  by  the  Romans,  p.  27. — 44.  The 
Tuscan  Order,  p.  30.— 45.  Egyptian  Style,  p.  30. — 46.  Building  in  General, 
p.  33.— 47.  Expression,  p.  35. — 48.  Durability,  p.  37.— 49.  Dwelling- 
Houses,  p.  37. — 5O.  Arranging  the  Stairs  and  Windows,  p.  42. — 51.  Prin- 
ciples of  Architecture,  p.  44.— 52.  Arrangement,  p.  44. — 53.  Ventilation,  p. 
45. — 54.  Stability,  p.  45. — 55.  Decoration,  p.  46.— 56.  Elementary  Parts  of 
a  Building,  p.  46. — 57.  The  Foundation,  p.  47. — 58.  The  Column,  or  Pillar, 
p.  47.— 59.  The  Wall,  p.  48. — 6O.  The  Reticulated  Walls,  p.  49.— 61.  The 
Lintel,  or  Beam,  p.  49. — 62.  The  Arch,  p.  50. — 63.  Hookc's  Theory  of  an 
Arch,  p.  50. — 64.  Gothic  Arches,  p.  51. — 65.  Arch  :  Definitions  ;  Principles, 
p.  52.— 66.  An  Arcade,  p.  52.— 67.  The  Vault,  p.  52.— 68.  The  Dome,  p.  53. 
—69.  The  Roof,  p.  54. 


614  CONTENTS. 

SECTION    II.— CONSTRUCTION. 

Art.  TO.  Construction  Essential,  p.  56. — 71.  Laws  of  Pressure,  p.  57. — 
72.  Parallelogram  of  Forces,  p.  59.— 73.  The  Resolution  of  Forces,  p.  59. — 
74.  Inclination  of  Supports  Unequal,  p.  60. — 75.  The  Strains  Exceed  the 
Weights,  p.  61. — 76.  Minimum  Thrust  of  Rafters,  p.  62. — 77.  Practical 
Method  of  Determining  Strains,  p.  62. — 78.  Horizontal  Thrust,  p.  63. — 79. 
Position  of  Supports,  p.  65. — 8O.  The  Composition  of  Forces,  p.  66. — 81. 
Another  Example,  p.  67. — 82.  Ties  and  Struts,  p.  68. — 83.  To  Distinguish 
Ties  from  Struts,  p.  69. — 84.  Another  Example,  p.  70. — 85.  Centre  of  Gravity, 
p.  7I._86.  Effect  of  the  Weight  of  Inclined  Beams,  p.  72. — 87.  Effect  of 
Load  on  Beam,  p.  74. — 88.  Effect  on  Bearings,  p.  75. — 89.  Weight-Strength, 
p.  76. — 9O.  Quality  of  Materials,  p.  76. — 91.  Manner  of  Resisting,  p.  77. — 
92.  Strength  and  Stiffness,  p.  78. — 93.  Experiments  :  Constants,  p.  78. — 94. 
Resistance  to  Compression,  p.  79. — 95.  Resistance  to  Tension,  p.  81. — 96. 
Resistance  to  Transverse  Strains,  p.  83. — 97.  Resistance  to  Compression,  p. 
85. — 98.  Compression  Transversely  to  the  Fibres,  p.  86. — 99.  The  Limit  of 
Weight,  p.  86.— 1OO.  Area  of  Post,  p.  86.— 1O1.  Rupture  by  Sliding,  p.  87. 
— 1O2.  The  Limit  of  Weight,  p.  87.— 1O3.  Area  of  Surface,  p.  88.— 1O4. 
Tenons  and  Splices,  p.  88. — HO5.  Stout  Posts,  p.  89.— IO6.  The  Limit  of 
Weight,  p.  89. — 1®7.  Area  of  Post,  p.  90. — 1O8.  Area  of  Round  Posts,  p. 
90.— 1O9.  Slender  Posts,  p.  91.— 11O.  The  Limit  of  Weight,  p.  91.— 111. 
Diameter  of  the  Post;  when  Round,  p.  92. — 112.  Side  of  Post:  when 
Square,  p.  93.— 113.  Thickness  of  a  Rectangular  Post,  p.  95. — 114.  Breadth 
of  a  Rectangular  Post,  p.  95.— 115.  Resistance  to  Tension,  p.  96. — 116. 
.The  Limit  of  Weight,  p.  96.— 117.  Sectional  Area,  p.  97.— 118.  Weight  of 
the  Suspending  Piece  Included,  p.  98. — 119.  Area  of  Suspending  Piece, 
p.  99. 

RESISTANCE   TO    TRANSVERSE    STRAINS. 

Art.  H2O.  Transverse  Strains:  Rupture,  p.  99. — H21L.  Location  of  Mor- 
tises, p.  100. — 122.  Transverse  Strains  :  Relation  of  Weight  to  Dimensions, 
p.  loi— 123.  Safe  Weight :  Load  at  Middle,  p.  103.— 124.  Breadth  of  Beam 
with  Safe  Load,  p.  104. — fl25.  Depth  of  Beam  with  Safe  Load,  p.  104. — 126. 
Safe  Load  at  any  Point,  p.  105. — 127.  Breadth  or  Depth  :  Load  at  any  Point, 
p.  106. — 128.  Weight  Uniformly  Distributed,  p.  107. — 129.  Breadth  or 
Depth:  Load  Uniformly  Distributed,  p.  108. — 13O.  Load  per  Foot  Super- 
ficial, p.  109. — 131.  Levers  :  Load  at  one  End,  p.  no. — 132.  Levers  :  Breadth 
or  Depth,  p.  in. — 133.  Deflection:  Relation  to  Weight,  p.  112. — 134.  De- 
flection: Relation  t<)  Dimensions,  p.  112. — 135.  Deflection  :  Weight  when  at 
Middle,  p.  114. — 136.  Deflection  :  Breadth  or  Depth,  Weight  at  Middle,  p. 
114.— 137.  Deflection  :  When  Weight  is  at  Middle,  p.  116.— 138.  Deflection  : 
Load  Uniformly  Distributed,  p.  116. — 139.  Deflection  :  Weight  when  Uni- 
formly Distributed,  p.  117.— I4O.  Deflection:  Breadth  or  Depth,  Load  Uni- 
formly Distributed,  p.  117. — 14i.  Deflection:  When  Weight  is  Uniformly 
Distributed,  p.  118. — 142.  Deflection  of  Lever,  p.  119. — 143.  Deflection  of 
a  Lever:  Load  at  End,  p.  120. — 144.  Deflection  of  a  Lever  :  Weight  when  at 
End,  p.  120. — 145.  Deflection  of  a  Lever  :  Breadth  or  Depth,  Load  at  End, 


CONTENTS.  615 

p.  i2i.— 146.  Deflection  of  Levers:  Weight  Uniformly  Distributed,  p.  121.— 
147.  Deflection  of  Levers  with  Uniformly  Distributed  Load,  p.  122.. — 148. 
Deflection  of  Levers:  Weight  when  Uniformly  Distributed,  p.  122. — 149. 
Deflection  of  Levers  :  Breadth  or  Depth,  Load  Uniformly  Distributed,  p.  122. 

CONSTRUCTION    IN    GENERAL. 

Art.  15O.  Construction :  Object  Clearly  Denned,  p.  123. — 151.  Floors 
Described,  p.  124. — 152.  Floor-Beams,  p.  125. — 153.  Floor-Beams  for  Dwell- 
ings, p.  127. — 154.  Floor-Beams  for  First-Class  Stores,  p.  128. — 155.  Floor- 
Beams  :  Distance  from  Centres,  p.  129. — 156.  Framed  Openings  for  Chimneys 
and  Stairs,  p.  130. — 157.  Breadth  of  Headers,  p.  130. — 15§.  Breadth  of 
Carriage-Beams,  p.  132. — 159.  Breadth  of  Carriage-Beams  Carrying  Two 
Sets  of  Tail-Beams,  p.  134. — 16O.  Breadth  of  Carriage-Beam  with  Well-Hole 
at  Middle,  p.  136. — 161.  Cross- Bridging,  or  Herring-Bone  Bridging,  p.  137. 
—162.  Bridging  :  Value  to  Resist  Concentrated  Loads,  p.  137. — 163.  Gird- 
ers, p.  140. — 164.  Girders :  Dimensions,  p.  141. 

FIRE-PROOF   TIMBER   FLOORS. 

Art.  165.  Solid  Timber  Floors,  p.  143. — 166.  Solid  Timber  Floors  for 
Dwellings  and  Assembly-Rooms,  p.  143. — 167.  Solid  Timber  Floors  for  First- 
Class  Stores,  p.  144. — 168.  Rolled-Iron  Beams,  p.  145. — 169.  Rolled-Iron 
Beams:  Dimensions;  Weight  at  Middle,  p.  146. — 17O.  Rolled-Iron  Beams: 
Deflection  when  Weight  is  at  Middle,  p.  147. — 171.  Rolled-Iron  Beams: 
Weight  when  at  Middle,  p.  148. — 172.  Rolled-Iron  Beams:  Weight  at  any 
Point,  p.  148. — 173.  Rolled-Iron  Beams:  Dimensions  ;  Weight  at  any  Point, 
p  ^g. — B.74.  Rolled-Iron  Beams  :  Dimensions  ;  Weight  Uniformly  Distrib- 
uted, p.  149. — 175.  Rolled-Iron  Beams  :  Deflection  ;  Weight  Uniformly  Dis- 
tributed, p.  150.  — 176.  Rolled-Iron  Beams  :  Weight  when  Uniformly  Distrib- 
uted, p.  151. — 177.  Rolled-Iron  Beams:  Floors  of  Dwellings  or  Assembly- 
Rooms,  p.  151. — 178.  Rolled-Iron  Beams  :  Floors  of  First-Class  Stores,  p. 
152. — 179.  Floor-Arches:  General  Considerations,  p.  153. — 18O.  Floor- 
Arches:  Tie-Rods;  Dwellings,  p.  153.  — 181.  Floor-Arches:  Tie-Rods; 
First  Class  Stores,  p.  153. 

TUBULAR    IRON    GIRDERS. 

Art.  182.  Tubular  Iron  Girders:  Description,  p.  154.— 183.  Tubular 
Iron  Girders  :  Area  of  Flanges  ;  Load  at  Middle,  p.  154. — 184.  Tubular  Iron 
Girders  :  Area  of  Flanges  ;  Load  at  any  Point,  p.  155. — 185.  Tubular  Iron 
Girders  :  Area  of  Flanges  ;  Load  Uniformly  Distributed,  p.  156. — 186.  Tu- 
bular Iron  Girders:  Shearing  Strain,  p.  157. — 187.  Tubular  Iron  Girders: 
Thickness  of  Web,  p.  158. — 188.  Tubular  Iron  Girders  for  Floors  of  Dwell- 
ings, Assembly-Rooms,  and  Office  Buildings,  p.  159.— 189.  Tubular  Iron 
Girders  for  Floors  of  First-Class  Stores,  p.  160. 

CAST-IRON    GIRDERS. 

Art.  19O.  Cast-Iron  Girders:  Inferior,  p.  161. — 191.  Cast-Iron  Girder: 
Load  at  Middle,  p.  161. — 192.  Cast-Iron  Girder:  Load  Uniformly  Distributed, 


6l6  CONTENTS. 

p.  163. — 193.  Cast-Iron  Bowstring  Girder,  p.  163. — 194.  Substitute   for  the 
Bowstring  Girder,  p.  163. 

FRAMED    GIRDERS. 

Art.  195.  Graphic  Representation  of  Strains,  p.  165. — 196.  Framed 
Girders,  p.  166. — 197.  Framed  Girder  and  Diagram  of  Forces,  p.  167. — 198. 
Framed  Girders  :  Load  on  Both  Chords,  p.  171. — 199.  Framed  Girders:  Di- 
mensions of  Parts,  p.  173. 

PARTITIONS. 

Art.  2OO.  Partitions,  p.  174. — 2O1.  Examples  of  Partitions,  p.  175. 

ROOFS. 

Art.  2O2.  Roofs,  p.  178.— 2O3.  Comparison  of  Roof-Trusses,  p.  178.— 
2O4.  Force  Diagram  :  Load  upon  Each  Support,  p.  179. — 2O5.  Force  Dia- 
gram for  Truss  in  Fig.  59,  p.  179. — 2O6.  Force  Diagram  for  Truss  in  Fig.  60, 
p.  180. — 2O7.  Force  Diagram  for  Truss  in  JFtg.Gi,  p.  181. — 2O8.  Force  Dia- 
gram for  Truss  in  Fig.  63,  p.  183. — 299.  Force  Diagram  for  Truss  in  Fig.  64, 
p.  184.— 21O.  Forc«  Diagram  for  Truss  in  Fig.  65,  p.  185.— 211.  Force  Dia- 
gram for  Truss  in  Fig.  66,  p.  186.— 212.  Roof-Truss  :  Effect  of  Elevating  the 
Tie-Beam,  p.  187. — 213.  Planning  a  Roof,  p.  188. — 214.  Load  upon  Roof- 
Truss,  p.  189. — 215.  Load  on  Roof  per  Superficial  Foot,  p.  189.— 216.  Load 
upon  Tie-Beam,  p.  190.— 217.  Roof  Weights  in  Detail,  p.  191.— 21§.  Load 
per  Foot  Horizontal,  p.  192. — 219.  Weight  of  Truss,  p.  192. — 22O.  Weight 
of  Snow  on  Roofs,  p.  193. — 221.  Effect  of  Wind  on  Roofs,  p.  193.— 222. 
Total  Load  per  Foot  Horizontal,  p.  197. — 223.  Strains  in  Roof  Timbers 
Computed,  p.  198. — 224.  Strains  in  Roof  Timbers  Shown  Geometrically,  p. 
199. — 225.  Application  of  the  Geometrical  System  of  Strains,  p.  202. — 226. 
Roof  Timbers  :  the  Tie-Beam,  p.  204.— 227.  The  Rafter,  p.  205.— 228.  The 
Braces,  p.  208. — 229.  The  Suspension  Rod,  p.  210. — 23O.  Roof-Beams, 
Jack-Rafters,  and  Purlins,  p.  211. — 231.  Five  Examples  of  Roofs,  p.  212. — 
232.  Roof-Truss  with  Elevated  Tie-Beam,  p.  214. — 233.  Hip-Roofs  :  Lines 
and  Bevels,  p.  215. — 234.  The  Backing  of  the  Hip-Rafter,  p.  216. 

DOMES. 

Art.  235.  Domes,  p.  216.— 236.  Ribbed  Dome,  p.  217.— 237.  Domes: 
Curve  of  Equilibrium,  p.  218. — 238.  Cubic  Parabola  Computed,  p.  219. — 
239.  Small  Domes  over  Stairways,  p.  220. — 24O.  Covering  for  a  Spherical 
Dome,  p.  221. — 241.  Polygonal  Dome  :  Form  of  Angle-Rib,  p.  223. 

BRIDGES. 

Art.  242.  Bridges,  p.  223.— 243.  Bridges  :  Built-Rib,  p.  224.— 244. 
Bridges  :  Framed  Rib,  p.  226. — 245.  Bridges :  Roadway,  p.  227. — 246. 
Bridges  :  Abutments,  p.  227. — 247.  Centres  for  Stone  Bridges,  p.  229. — 248. 
Arch  Stones  :  Joints,  p.  223. 

JOINTS. 

Art.  249.  Timber  Joints,  p.  234. 


CONTENTS.  6i; 

SECTION  III.— STAIRS. 

Art.  250.  Stairs  :  General  Requirements,  p.  240. — 251.  The  Grade  of 
Stairs,  p.  241.— 252.  Pitch-Board  :  Relation  of  Rise  to  Tread,  p.  242. — 253. 
Dimensions  of  the  Pitch-Board,  p.  247. — 254.  The  String  of  a  Stairs,  p.  247. 
— 255.  Step  and  Riser  Connection,  p.  248. 

PLATFORM    STAIRS 

Art.  256.  Platform  Stairs :  the  Cylinder,  p.  248.— 257.  Form  of  Lower 
Edge  of  Cylinder,  p.  249. — 25§.  Position  of  the  Balusters,  p.  250. — 259.  Wind- 
ing Stairs,  p.  251. — 260.  Regular  Winding  Stairs,  p.  251.— 261.  Winding 
Stairs :  Shape  and  Position  of  Timbers,  p.  252. — 262.  Winding  Stairs  with 
Flyers:  Grade  of  Front  String,  p.  253. 

HAND-RAILING. 

Art.  263.  Hand-Railing  for  Stairs,  p.  256.— 264.  Hand-Railing :  Defini- 
tions ;  Planes  and  Solids, p.  257. — 265.  Hand-Railing:  Preliminary  Consider- 
ations, p.  258. — 266.  A  Prism  Cut  by  an  Oblique  Plane,  p.  259. — 267.  Form 
of  Top  of  Prism,  p.  259.— 26§.  Face-Mould  for  Hand-Railing  of  Platform 
Stairs,  p.  264. — 269.  More  Simple  Method  for  Hand-Rail  to  Platform  Stairs, 
p.  267. — 27O.  Hand-Railing  for  a  Larger  Cylinder,  p.  271. — 271.  Face- 
Mould  without  Canting  the  Plank,  p.  272. — 272.  Railing  for  Platform  Stairs 
where  the  Rake  meets  the  Level,  p.  272. — 273.  Application  of  Face-Moulds 
to  Plank,  p.  273. — 274.  Face-Moulds  for  Moulded  Rails  upon  Platform 
Stairs,  p.  274. — 275.  Application  of  Face-Moulds  to  Plank,  p.  275. — 276. 
Hand-Railing  for  Circular  Stairs,  p.  278. — 277.  Face-Moulds  for  Circular 
Stairs,  p.  282.— 27§.  Face-Moulds,  for  Circular  Stairs,  p.  285—279.  Face- 
Moulds  for  Circular  Stairs,  again,  p.  287. — 280.  Hand-Railing  for  Winding 
Stairs,  p.  289.— 2§1.  Face-Moulds  for  Windjng  Stairs,  p.  290.— 2§2.  Face- 
Moulds  for  Winding  Stairs,  again,  p.  293. — 283.  Face-Moulds  :  Test  of  Accu- 
racy, p.  295.— 284.  Application  of  the  Face-Mould,  p.  297. — 285.  Face-Mould 
Curves  are  Elliptical,  p.  301. — 286.  Face-Moulds  for  Round  Rails,  p.  303. — 
287.  Position  of  the  Butt  Joint,  p.  303.— 288.  Scrolls  for  Hand-Rails:  Gen- 
eral Rule  for  Size  and  Position  of  the  Regulating  Square,  p.  308. — 289.  Cen- 
tres in  Regulating  Square,  p.  308. — 29O.  Scroll  for  Hand-Rail  Over  Curtail 
Step,  p.  309.— 291.  Scroll  for  Curtail  Step,  p.  310.— 292.  Position  of  Balus- 
ters Under  Scroll,  p.  310— 293.  Falling-Mould  for  Raking  Part  of  Scroll,  p. 
310. — 294.  Face-Mould  for  the  Scroll,  p.  311. — 295.  Form  of  Newel-Cap 
from  a  Section  of  the  Rail,  p.  312.— 296.  Boring  for  Balusters  in  a  Round  Rail 
before  it  is  Rounded,  p.  313. 

SPLAYED    WORK. 

Art.  297.  The  Bevels  in  Splayed  Work,  p.  314. 

SECTION    IV.  — DOORS   AND   WINDOWS. 

DOORS. 

Art.  298.  General  Requirements,  p.  315. — 299.  The  Proportion  between 
Width  and  Height,  p.  315.— 3OO.  Panels,  p.  316.— 3O1.  Trimmings,  p.  317. 
— 302.  Hanging  Doors,  p.  317. 


6l8  CONTENTS. 

WINDOWS. 

Art.  3O3.  Requirements  for  Light,  p.  317. — 3O4.  Window  Frames,  p. 
318.— 3O5.  Inside  Shutters,  p.  319.— 3O6.  Proportion:  Width  and  Height, 
p.  319.— 3O7.  Circular  Heads,  p.  320.— 3O8.  Form  of  Soffit  for  Circular  Win- 
dow Heads,  p.  321. 

SECTION   V.— MOULDINGS  AND  CORNICES. 

MOULDINGS. 

Art.  309.  Mouldings,  p.  323. — 31 0.  Characteristics  of  Mouldings,  p. 
324. — 311.  A  Profile,  p.  326. — 312.  The  Grecian  Torus  and  Scotia,  p.  326. — 
313.  The  Grecian  Echinus,  p.  327. — 314.  The  Grecian  Cavetto,  p.  327. — 
315.  The  Grecian  Cyma- Recta,  p.  327. — 316.  The  Grecian  Cyma-Reversa, 
p.  328.— 3S.7.  Roman  Mouldings,  p.  329. — 318.  Modern  Mouldings,  p.  331. 

CORNICES. 

Art.  3H9.  Designs  for  Cornices,  p.  335. — 32O.  Eave  Cornices  Propor- 
tioned to  Height  of  Building,  p.  335. — 321.  Cornice  Proportioned  to  a  given 
Cornice,  p.  342. — 322.  Angle  Bracket  in  a  Built  Cornice,  p.  343. — 323.  Rak- 
ing Mouldings  Matched  with  Level  Returns,  p.  344. 


PART  II. 


SECTION   VI.-GEOMETRY. 

Art.  324.  Mathematics  Essential,  p.  347. — 325.  Elementary  Geometry, 
p.  347. — 326.  Definition— Right  Angles,  p.  348. — 327.  Definition — Degrees 
in  a  Circle,  p.  348. — 328.  Definition — Measure  of  an  Angle,  p.  348.— 329. 
Corollary — Degrees  in  a  Right  Angle,  p.  348.— 33O.  Definition — Equal 
Angles,  p.  349. — 331.  Axiom — Equal  Angles,  p.  349. — 332.  Definition — 
Obtuse  and  Acute  Angles,  p.  349. — 333.  Axiom — Right  Angles,  p.  349. — 
334.  Corollary — Two  Right  Angles,  p.  349. — 335.  Corollary — Four  Right 
Angles,  p.  349. — 336.  Proposition — Equal  Angles,  p.  350.— 337.  Propo- 
sition— Equal  Triangles,  p.  350. — 338.  Proposition — Angles  in  Isosceles 
Triangle,  p.  351. — 339.  Proposition — Diagonal  of  Parallelogram,  p.  351- — 
34O.  Proposition — Equal  Parallelograms,  p.  352. — 341.  Proposition — Paral- 
lelograms Standing  on  the  Same  Base,  p.  352. — 342.  Corollary — Parallelo- 
gram and  Triangle,  p.  353. — 343.  Proposition — Triangle  Equal  to  Quadrangle, 
p.  353. — 344.  Proposition — Opposite  Angles  Equal,  p.  354. — 345.  Proposi- 
tion— Three  Angles  of  Triangle  Equal  to  Two  Right  Angles,  p.  354. — 346. 
Corollary— Right  Angle  in  Triangle,  p.  354.— 347.  Corollary— Half  a  Right 


CONTENTS.  619 

Angle,  p.  355.— 348.  Corollary— Right  Angle  in  a  Triangle,  p.  355.— 349. 
Corollary — Two  Angles  Equal  to  Right  Angle,  p.  355.—  35O.  Corollary — Two 
Thirds  of  a  Right  Angle,  p.  355. — 351.  Corollary — Equilateral  Triangle,  p.  355. 
— 352.  Proposition — Right  Angle  in  Semi-circle,  p.  355. — 353.  Proposition—- 
The  Square  of  the  Hypothenuse  Equal  to  the  Squares  of  the  Sides,  p.  355. — 
354.  Proposition — Equilateral  Octagon,  p.  357. — 355.  Proposition — Angle 
at  the  Circumference  of  a  Circle,  p.  358.— 356.  Proposition — Equal  Chords 
give  Equal  Angles,  p.  358. — 357.  Corollary  of  Equal  Chords,  p.  359.— 35§. 
Proposition— Angle  Formed  by  a  Chord  and  Tangent,  p.  359. — 359.  Propo- 
sition— Areas  of  Parallelograms,  p.  360. — 360.  Proposition — Triangles  ot 
Equal  Altitude,  p.  361. — 361.  Proposition— Homologous  Triangles,  p.  362. — 
362.  Proposition — Parallelograms  of  Chords,  p.  363. — 363.  Proposition — • 
Sides  of  Quadrangle,  p.  364. 

SECTION    VII.— RATIO,  OR   PROPORTION. 

Art.  364.  Merchandise,  p.  366.— 365.  The  Rule  of  Three,  p.  366.— 
366.  Couples:  Antecedent,  Consequent,  p.  367. — 367.  Equal  Couples  :  an 
Equation,  p.  367. — 36§.  Equality  of  Ratios,  p.  367. — 369.  Equals  Multiplied 
by  Equals  Give  Equals,  p.  367. — 37O.  Multiplying  an  Equation,  p.  368. — 371. 
Multiplying  and  Dividing  one  Member  of  an  Equation  :  Cancelling,  p.  368. — 
372.  Transferring  a  Factor,  p.  369. — 373.  Equality  of  Product :  Means  and 
Extremes,  p.  369. — 374.  Homologous  Triangles  Proportionate,  p.  370. — 
375.  The  Steelyard,  p.  371.— 376.  The  Lever  Exemplified  by  the  Steelyard, 
p.  372. — 377.  The  Lever  Principle  Demonstrated,  p.  375. — 378.  Any  One  or 
Four  Proportionals  may  be  Found,  p.  377. 

SECTION   VIII.— FRACTIONS. 

Art.  379.  A  Fraction  Defined,  p.  378.— 38O.  Graphical  Representation 
of  Fractions  :  Effect  of  Multiplication,  p.  378. — 381.  Form  of  Fraction 
Changed  by  Division,  p.  380. — 382.  Improper  Fractions,  p.  380. — 383.  Re- 
duction of  Mixed  Numbers  to  Fractions,  p.  381. — 384.  Division  Indicated  by 
the  Factors  put  as  a  Fraction,  p.  381.— 385.  Addition  of  Fractions  having  Like 
Denominators,  p.  382. — 386.  Subtraction  of  Fractions  of  Like  Denominators, 
p>  383. — 387.  Dissimilar  Denominators  Equalized,  p.  383. — 388.  Reduction 
of  Fractions  to  their  Lowest  Terms,  p.  384. — 389.  Least  Common  Denomina- 
tor, p.  384. — 390.  Least  Common  Denominator  Again,  p.  385. — 391.  Frac- 
tions Multiplied  Graphically,  p.  386.— 392.  Fractions  Multiplied  Graphically 
Again,  p.  387.— 393.  Rule  for  Multiplication  of  Fractions,  and  Example,  p. 
387.—  394.  Fractions  Divided  Graphically,  p.  388.— 395.  Rule  for  Division 
of  Fractions,  p.  389. 

SECTION    IX.— ALGEBRA. 

Art.  396.  Algebra  Defined,  p.  392.— 397.  Example:  Application,  p. 
393> — 398.  Algebra  Useful  in  Constructing  Rules,  p.  394. — 399.  Algebraic 
Rules  are  General,  p.  394.— 4OO.  Symbols  Chosen  at  Pleasure,  p.  395.— 4O1. 
Arithmetical  Processes  Indicated  by  Signs,  p.  396. — 4O2.  Examples  in  Addi- 


C2O  CONTENTS. 

tion  and  Subtraction  :  Cancelling,  p.  398. — 403.  Transferring  a  Symbol  to  the 
Opposite  Member,  p.  399. — 4O4.  Signs  of  Symbols  to  be  Changed  when  they 
are  to  be  Subtracted,  p.  400. — 4O5.  Algebraic  Fractions,  Added  and  Sub- 
tracted, p.  403.— 4O6.  The  Least  Common  Denominator.,  p.  404.—  4O7.  Alge- 
braic Fractions  Subtracted,  p.  405. — 4O§.  Graphical  Representation  of  Multi- 
plication, p.  408. — 4O9.  Graphical  Multiplication  :  Three  Factors,  p.  408. — 
41O.  Graphic  Representation  :  Two  and  Three  Factors,  p.  409. — 411.  Graph- 
ical Multiplication  of  a  Binomial,  p.  409. — 412.  Graphical  Squaring  of  a 
Binomial,  p.  410.— 413.  Graphical  Squaring  of  the  Difference  of  Two  Fac- 
tors, p.  412. — 414.  Graphical  Product  of  the  Sum  and  Difference  of  Two 
Quantities,  p.  413. — 415.  Plus  and  Minus  Signs  in  Multiplication,  p.  415, — 
416.  Equality  of  Squares  on  Hypothenuse  and  Sides  of  Right-Angled  Tri- 
angle, p.  416. — 417*.  Division  the  Reverse  of  Multiplication,  p.  418. — 418. 
Division:  Statement  of  Quotient,  p.  419. — 419.  Division:  Reduction,  p.  419. 
—  420.  Proportionals  :  Analysis,  p.  421. — 421.  Raising  a  Quantity  to  any 
Power,  p.  423. — 422.  Quantities  with  Negative  Exponents,  p.  423.— 423. 
Addition  and  Subtraction  of  Exponential  Quantities,  p.  424. — 424.  Multipli- 
cation of  Exponential  Quantities,  p.  424. — 425.  Division  of  Exponential 
Quantities,  p.  424. — 426.  Extraction  of  Radicals,  p.  425. — 427.  Logarithms, 
p.  425. — 428.  Completing  the  Square  of  a  Binomial,  p.  429. 

PROGRESSION. 

Art.  429.  Arithmetical  Progression,  p.  432. — 43O.  Geometrical  ProgreS' 
sion,  p.  435. 

SECTION   X.--POLYGONS. 

Art.  431.  Relation  of  Sum  and  Difference  of  Two  Lines,  p.  439. — 432. 

Perpendicular,  in  Triangle  of  Known  Sides,  p.  440. — 433.  Trigon  :  Radius  of 
Circumscribed  and  Inscribed  Circles  :  Area,  p.  443. — 434.  Tetragon  :  Radius 
of  Circumscribed  and  Inscribed  Circles:  Area,  p.  446. — 435.  Hexagon  :  Ra- 
dius ot  Circumscribed  and  Inscribed  Circles :  Area,  p.  447. — 436.  Octagon  : 
Radius  of  Circumscribed  and  Inscribed  Circles  :  Area,  p.  449. — 437.  Dodec- 
agon: Radius  of  Circumscribed  and  Inscribed  Circles:  Area,  p.  452. — 438. 
Hecadecagon  :  Radius  of  Circumscribed  and  Inscribed  Circles  :  Area,  p.  455. 
— 439.  Polygons  :  Radius  of  Circumscribed  and  Inscribed  Circles  :  Area,  p. 
460. — 440.  Polygons  :  Their  Angles,  p.  462. — 441.  Pentagon:  Radius  of  the 
Circumscribed  and  Inscribed  Circles:  Area,  p.  463. — 442.  Polygons:  Table 
of  Constant  Multipliers,  p.  465. 


SECTION    XL— THE  CIRCLE. 

Art.  443.  Circles  :  Diameter  and  Perpendicular  :  Mean  Proportional,  p. 
468. — 444.  Circle  :  Radius  from  Given  Chord  and  Versed  Sine,  p.  469. — 
445.  Circle :  Segment  from  Ordinates,  p.  470. — 446.  Circle :  Relation  of 
Diameter  to  Circumference,  p.  472. — 447.  Circle  :  Length  of  an  Arc,  p.  475. 
— 448.  Circle  :  Area,  p.  475. — 449.  Circle:  Area  of  a  Sector,  p.  476. — 45O. 
Circle  :  Area  of  a  Segment,  p.  477. 


CONTENTS.  621 

SECTION   XII.— THE  ELLIPSE. 

Art.  451.  Ellipse:  Definitions,  p.  481. — 452.  Ellipse:  Equations  to  the 
Curve,  p.  482. — 453.  Ellipse  :  Relation  of  Axis  to  Abscissas  of  Axes,  p.  484. 
— 454.  Ellipse :  Relation  of  Parameter  and  Axes,  p.  485. — 455.  Ellipse  : 
Relation  of  Tangent  to  the  Axes,  p.  485. — 456.  Ellipse:  Relation  of  Tangent 
with  the  Foci,  p.  487. — 457.  Ellipse  :  Relation  of  Axes  to  Conjugate  Diam- 
eters, p.  487.— 458.  Ellipse  :  Area,  p.  488.— 459.  Ellipse  :  Practical  Sugges- 
tions, p.  489. 

SECTION   XIII.— THE   PARABOLA. 

Art.  46O.  Parabola  :  Definitions,  p.  492. — 461.  Parabola :  Equation  to 
the  Curve,  p.  493. — 462.  Parabola  :  Tangent,  p.  493. — 463.  Parabola  :  Sub- 
tangent,  p.  496. — 464.  Parabola :  Normal  and  Subnormal,  p.  496. — 465. 
Parabola  :  Diameters,  p.  497. — 466.  Parabola  :  Elements,  p.  499. — 467. 
Parabola  :  Described  Mechanically,  p.  500. — 468.  Parabola  :  Described  from 
Points,  p.  502. — 469.  Parabola  :  Described  from  Arcs,  p.  503. — 47O.  Para- 
bola :  Described  from  Ordinates,  p.  504. — 471.  Parabola :  Described  from 
Diameters,  p.  507. — 472.  Parabola  :  Area,  p.  509. 

SECTION  XIV.— TRIGONOMETRY. 

Art.  473.  Right-Angled  Triangles:  The  Sides,  p.  510.— 474.  Right- 
Angled  Triangles :  Trigonometrical  Tables,  p.  512. — 475*  Right-Angled 
Triangles:  Trigonometrical  Value  of  Sides,  p.  516. — 476.  Oblique- Angled 
Triangles:  Sines  and  Sides,  p.  519. — 477.  Oblique-Angled  Triangles :  First 
Class,  p.  520. — 478.  Oblique-Angled  Triangles:  Second  Class,  p.  522. — 
479.  Oblique-Angled  Triangles  :  Sum  and  Difference  of  Two  Angles,  p.  523. 
— 48O.  Oblique-Angled  Triangles :  Third  Class,  p.  526. — 481.  Oblique- 
Angled  Triangles  :  Fourth  Class,  p.  528. — 482*  Trigonometrical  Formulae  : 
Right-Angled  Triangles,  p.  530. — 483.  Trigonometrical  Formula;  :  First 
Class,  Oblique,  p.  531. — 484.  Trigonometrical  Formulae:  Second  Class, 
Oblique,  p.  532. — 485.  Trigonometrical  Formulae  :  Third  Class,  Oblique,  p. 
534.— 486.  Trigonometrical  Formulas  :  Fourth  Class,  Oblique,  p.  534. 


SECTION  XV.— DRAWING 

Art.  487.  General    Remarks,  p.  536.— 488.  Articles  Required,  p.  536.— 
489.  The  Drawing-Board,  p.  536.— 49O.  Drawing- Paper,  p.  537.— 491.  To 

Secure  the  Paper  to  the  Board,  p.  537.— 492.  The  T-Square,  p.  539.— 493. 
The  Set-Square,  p.  539. — 494.  The  Rulers,  p.  540.— 495.  The  Instruments, 
p.  540.— 496.  The  Scale  of  Equal  Parts,  p.  540.— 497.  The  Use  of  the  Set- 
Square,  p..  541. — 498.  Directions  for  Drawing,  p.  542. 


622  CONTENTS. 

SECTION    XVI.— PRACTICAL  GEOMETRY. 
Art.  499.  Definitions  of  Various  Terms,  p.  544. 

PROBLEMS. 

RIGHT   LINES   AND    ANGLES. 

Art.  5OO.  To  Bisect  a  Line,  p.  549.— 5O1.  To  Erect  a  Perpendicular,  p. 
550.— 5O2.  To  let  Fall  a  Perpendicular,  p.  551.— 5O3.  To  Erect  a  Perpen- 
dicular at  the  End  of  a  Line,  p.  551. — 5O4.  To  let  Fall  a  Perpendicular  near 
the  End  of  a  Line,  p.  553. — 5O5.  To  Make  an  Angle  Equal  to  a  Given  Angle, 
p.  553. — 5O6.  To  Bisect  an  Angle,  p.  554. — 5O7.  To  Trisect  a  Right  Angle, 
p.  554. — 5O8.  Through  a  Given  Point  to  Draw  a  Line  Parallel  to  a  Given 
Line,  p.  555. — 509.  To  Divide  a  Given  Line  into  any  Number  of  Equal  Parts, 
P- 555- 

THE   CIRCLE. 

Art.  5IO.  To  Find  the  Centre  of  a  Circle,  p.  556. — 511.  At  a  Given 
Point  in  a  Circle  to  Draw  a  Tangent  thereto,  p.  557. — 512.  The  Same,  with- 
out making  use  of  the  Centre  of  the  Circle,  p.  557. — 513.  A  Circle  and  a 
Tangent  Given,  to  Find  the  Point  of  Contact,  p.  558. — 514.  Through  any 
Three  Points  not  in  a  Straight  Line  to  Draw  a  Circle,  p.  559.— 515.  Three 
Points  not  in  a  Straight  Line  being  Given,  to  Find  a  Fourth  that  Shall,  with 
the  Three,  Lie  in  the  Circumference  of  a  Circle,  p.  559. — 516.  To  Describe 
a  Segment  of  a  Circle  by  a  Set-Triangle,  p.  560. — 517.  To  Find  the  Radius  of 
an  Ate  of  a  Circle  when  the  Chord  and  Versed  Sine  are  Given,  p.  561. — 518. 
To  Find  the  Versed  Sine  of  an  Arc  of  a  Circle  when  the  Radius  and  Chord 
are  Given,  p.  561. — 519.  To  Describe  the  Segment  of  a  Circle  by  Intersection 
of  Lines,  p.  562. — 52O.  Ordinates,  p.  563. — 521.  In  a  Given  Angle  to  De- 
scribe a  Tanged  Curve,  p.  565. — 522.  To  Describe  a  Circle  within  any  Given 
Triangle,  so  that  the  Sides  of  the  Triangle  shall  be  Tangical,  p.  566.— 523. 
About  a  Given  Circle  to  Describe  an  Equilateral  Triangle,  p.  566. — 524.  To 
Find  a  Right  Line  nearly  Equal  to  the  Circumference  of  a  Circle,  p.  566. 

POLYGONS,   ETC. 

Art.  525.  Upon  a  Given  Line  to  Construct  an  Equilateral  Triangle,  p. 
568.— 526.  To  Describe  an  Equi'lateral  Rectangle,  or  Square,  p.  568. — 527. 
Within  a  Given  Circle  to  Inscribe  an  Equilateral  Triangle,  Hexagon,  or  Dodec- 
agon, p.  569. — 52§.  Within  a  Square  to  Inscribe  an  Octagon,  p.  570. — 529. 
To  Find  the  Side  of  a  Buttressed  Octagon,  p.  571. — 5ilO.  Within  a  Given 
Circle  to  Inscribe  any  Regular  Polygon,  p.  572. — 531.  Upon  a  Given  Line  to 
Describe  any  Regular  Polygon,  p.  573. — 532.  To  Construct  a  Triangle  whose 
Sides  shall  be  severally  Equal  to  Three  Given  Lines,  p.  575. — 533.  To  Con- 
struct a  Figure  Equal  to  a  Given  Right-lined  Figure,  p.  575. — 534.  To  Make 
a  Parallelogram  Equal  to  a  Given  Triangle,  p.  576. — 535.  A  Parallelogram 
being  Given,  to  Construct  Another  Equal  to  it,  and  Having  a  Side  Ecjual  to  a 


CONTENTS.  623 

Given  Line,  p.  576. — 536.  To  Make  a  Square  Equal  to  two  or  more  Given 
Squares,  p.  577.— 537.  To  Make  a  Circle  Equal  to  two  Given  Circles,  p.  580. 
— 53§.  To  Construct  a  Square  Equal  to  a  Given  Rectangle,  p.  581. — 539.  To 
Form  a  Square  Equal  to  a  Given  Triangle,  p.  582. — 54O.  Two  Right  Lines 
being  Given,  to  Find  a  Third  Proportional  thereto,  p.  582. — 541.  Three  Right 
Lines  being  Given,  to  Find  a  Fourth  Proportional  thereto,  p.  583. — 542.  A 
Line  with  Certain  Divisions  being  Given,  to  Divide  Another,  Longer  or 
Shorter,  Given  Line  in  the  Same  Proportion,  p.  583. — 543.  Between  Two 
Given  Right  Lines  to  Find  a  Mean  Proportional,  p.  584. 


CONIC    SECTIONS. 

Art.  544.  Definitions,  p.  584. — 545.  To  Find  the  Axes  of  the  Ellipsis, 
p.  585.— 546.  To  Find  the  Axis  and  Base  of  the  Parabola,  p.  585.— 547.  To 
Find  the  Height,  Base,  and  Transverse  Axis  of  an  Hyperbola,  p.  585. — 54§. 
The  Axes  being  Given,  to  Find  the  Foci,  and  to  Describe  an  Ellipsis  with  a 
String,  p.  586. — 549.  The  Axes  being  Given,  to  Describe  an  Ellipsis  with  a 
Trammel,  p.  586. — 55O.  To  Describe  an  Ellipsis  by  Ordinates,  p.  588.— 551. 
To  Describe  an  Ellipsis  by  Intersection  of  Lines,  p.  588. — 552.  To  Describe 
an  Ellipsis  by  Intersecting  Arcs,  p.  590. — 553.  To  Describe;  a  Figure  Nearly 
in  the  Shape  of  an  Ellipsis  by  a  Pair  of  Compasses,  p.  591.— 554.  To  Draw 
an  Oval  in  the  Proportion  Seven  by  Nine,  p.  591. — 555.  To  Draw  a  Tangent 
to  an  Ellipsis,  p.  592. — 556.  An  Ellipsis  with  a  Tangent  Given,  to  Detect  the 
Point  of  Contact,  p.  593. — 557.  A  Diameter  of  an  Ellipsis  Given,  to  Find  its 
Conjugate,  p.  593. — 558.  Any  Diameter  and  its  Conjugate  being  Given,  to 
Ascertain  the  Two  Axes,  and  thence  to  Describe  the  Ellipsis,  p.  593. — 559. 
To  Describe  an  Ellipsis,  whence  Axes  shall  be  Proportionate  to  the  Axes  of 
a  Larger  or  Smaller  Given  One,  p.  594.— 56O.  To  Describe  a  Parabola  by 
Intersection  of  Lines,  p.  594. — 561.  To  Describe  an  Hyperbola  by  Intersec- 
tion of  Lines,  p.  595. 


SECTION    XVII.— SHADOWS. 

Art.  562.  The  Art  of  Drawing,  p.  596.— 563.  The  Inclination  of  the  Line 
of  Shadow,  p.  596. — 564.  To  Find  the  Line  of  Shadow  on  Mouldings  and 
other  Horizontally  Straight  Projections,  p.  597. — 565.  To  Find  the  Line  of 
Shadow  Cast  by  a  Shelf,  p.  598.— 566.  To  Find  the  Shadow  Cast  by  a  Shelf 
which  is  Wider  at  one  End  than  at  the  Other,  p.  599. — 567.  To  Find  the 
Shadow  of  a  Shelf  having  one  End  Acute  or  Obtuse  Angled,  p.  600. — 56§. 
To  Find  the  Shadow  Cast  by  an  Inclined  Shelf,  p.  600. — 569.  To  Find  the 
Shadow  Cast  by  a  Shelf  inclined  in  its  Vertical  Section  either  Upward  or 
Downward,  p.  601. — 57O.  To  Find  the  Shadow  of  a  Shelf  having  its  Front 
Edge  or  End  Curved  on  the  Plan,  p.  602. — 571.  To  Find  the  Shadow  of  a 
Shelf  Curved  in  the  Elevation,  p.  602.— 572.  To  Find  the  Shadow  Cast  upon 
a  Cylindrical  Wall  by  a  Projection  of  any  Kind,  p.  603.— 573.  To  Find  the 
Shadow  Cast  by  a  Shelf  upon  an  Inclined  Wall.  p.  603.— 574.  To  Find  the 
Shadow  of  a  Projecting  Horizontal  Beam,  p  604.— 575.  To  Find  the  Shadow 


624  CONTENTS. 

in  a  Recess,  p.  604.— 576.  To  Find  the  Shadow  in  a  Recess,  when  the  Face  of 
the  Wall  is  Inclined,  and  the  Back  of  the  Recess  is  Vertical,  p.  604. — 577. 
To  Find  the  Shadow  in  a  Fireplace,  p.  605. — 578.  To  Find  the  Shadow  of  a 
Moulded  Window-Lintel,  p.  606. — 579.  To  Find  the  Shadow  Cast  by  the 
Nosing  of  a  Step,  p.  606. — 58O.  To  Find  the  Shadow  Thrown  by  a  Pedestal 
upon  Steps,  p.  6c6. — 5§1.  To  Find  the  Shadow  Thrown  on  a  Column  by  a 
Square  Abacus,  p.  607. — 582.  To  Find  the  Shadow  Thrown  on  a  Column  by 
a  Circular  Abacus,  p.  608. — 583.  To  Find  the  Shadows  on  the  Capital  of  a 
Column,  p.  609.— 584.  To  Find  the  Shadow  Thrown  on  a  Vertical  Wall  by  a 
Column  and  Entablature  Standing  in  Advance  of  said  Wall,  p.  611. — 585* 
Shadows  on  a  Cornice,  p.  611. — 596.  Reflected  Light,  p.  611. 


AMERICAN   HOUSE   CARPENTER. 


APPENDIX. 


UNIVERSITY 


CONTENTS. 


PAGE. 

GLOSSARY 627 

TABLE  OF  SQUARES,  CUBES,  AND  ROOTS 638 

RULES  FOR  THE  REDUCTION  OF  DECIMALS 647 

TABLE  OF  CIRCLES 649 

TABLE  SHOWING  THE  CAPACTTY  OF  WELLS,  CISTERNS,  ETC 653 

TABLE  OF  THE  WEIGHTS  OF  MATERIALS 654 


GLOSSARY. 


Terms  not  found  here  can  be  found  in  the  lists  of  definitions  in  other  parts  of  this  book,  or  in 

common  dictionaries. 


Abacus. — The  uppermost  member  of  a  capital. 

Abattoir. — A  slaughter-house. 

Abbey. — The  residence  of  an  abbot  or  abbess. 

Abutment. — That  part  of  a  pier  from  which  the  arch  springs. 

Acanthus. — A  plant  called  in  English  bear' s-breech.  Its  leaves  are  employed 
for  decorating  the  Corinthian  and  the  Composite  capitals. 

Acropolis. — The  highest  part  of  a  city  ;  generally  the  citadel. 

Acroteria. — The  small  pedestals  placed  on  the  extremities  and  apex  of  a 
pediment,  originally  intended  as  a  base  for  sculpture. 

Aisle. — Passage  to  and  from  the  pews  of  a  church.  In  Gothic  architecture, 
the  lean-to  wings  on  the  sides  of  the  nave. 

Alcove. — Part  of  a  chamber  separated  by  an  estrade,  or  partition  of  columns. 
Recess  with  seats,  etc.,  in  gardens. 

Altar. — A  pedestal  whereon  sacrifice  was  offered.  In  modern  churches,  the 
area  within  the  railing  in  front  of  the  pulpit. 

Alto-relievo. — High  relief ;  sculpture  projecting  from  a  surface  so  as  to  appear 
nearly  isolated. 

Amphitheatre. — A  double  theatre,  employed  by  the  ancients  for  the  exhibi- 
tion of  gladiatorial  fights  and  other  shows. 

Ancones. — Trusses  employed  as  an  apparent  support  to  a  cornice  upon  the 
flanks  of  the  architrave. 

Annulet.— A  small  square  moulding  used  to  separate  others  ;  the  fillets  in 
the  Doric  capital  under  the  ovolo,  and  those  which  separate  the  flutings  of  col- 
umns, are  known  by  this  term. 

Antce. — A  pilaster  attached  to  a  wall. 

Apiary.— A  place  for  keeping  beehives. 

Arabesque. — A  building  after  the  Arabian  style. 

Areostyle.—  An  intercolumniation  of  from  four  to  five  diameters. 

Arcade. — A  series  of  arches. 

Arch. An  arrangement  of  stones  or  other  material  in  a  curvilinear  form,  so 

as  to  perform  the  office  of  a  lintel  and  carry  superincumbent  weights. 

Architrave.— That  part  of  the  entablature  which  rests  upon  the  capital  of  a 
column,  and  is  beneath  the  frieze.  The  casing  and  mouldings  about  a  door  or 
window. 

Archivolt.—The  ceiling  of  a  vault  ;  the  under  surface  of  an  arch. 

Area.— Superficial  measurement.  An  open  space,  below  the  level  of  the 
ground,  in  front  of  basement  windows. 


628  APPENDIX. 

Arsenal. — A  public  establishment  for  the  deposition  of  arms  and  warlike 
stores. 

Astragal. — A  small  moulding  consisting  of  a  half-round  with  a  fillet  on  each 
side. 

Attic.— A.  low  story  erected  over  an  order  of  architecture.  A  low  additional 
story  immediately  under  the  roof  of  a  building. 

Aviary. — A  place  for  keeping  and  breeding  birds. 

Balcony. — An  open  gallery  projecting  from  the  front  of  a  building. 

Baluster. — A  small  pillar  or  pilaster  supporting  a  rail. 

Balustrade. — A  series  of  balusters  connected  by  a  rail. 

Barge-course. — That  part  of  the  covering  which  projects  over  the  gable  of  a 
building. 

Base. — The  lowest  part  of  a  wall,  column,  etc. 

Basement-story. — That  which  is  immediately  under  the  principal  story,  and 
included  within  the  foundation  of  the  building. 

Basso-relievo. —  Low  relief ;  sculptured  figures  projecting  from  a  surface  one 
half  their  thickness  or  less.  See  Alto-relievo. 

Battering. — See  Talus. 

Battlement. — Indentations  on  the  top  of  a  wall  or  parapet. 

Bay-window. — A  window  projecting  in  two  or  more  planes,  and  not  form- 
ing the  segment  of  a  circle. 

Bazaar. — A  species  of  mart  or  exchange  for  the  sale  of  various  articles  of 
merchandise. 

Bead. — A  circular  moulding. 

Bed-mouldings. — Those  mouldings  which  are  between  the  corona  and  the 
frieze. 

Belfry. — That  part  of  the  steeple  in  which  the  bells  are  hung ;  anciently 
called  campanile. 

Belvedere. — An  ornamental  turret  or  observatory  commanding  a  pleasant 
prospect. 

Bow-window. — A  window  projecting  in  curved  lines. 

Bressummer. — A  beam  or  iron  tie  supporting  a  wall  over  a  gateway  or  other 
opening. 

Brick-nogging. — The  brickwork  between  studs  of  partitions. 

Buttress. — A  projection  from  a  wall  to  give  additional  strength. 

Cable. — A  cylindrical  moulding  placed  in  flutes  at  the  lower  part  of  the  col- 
umn. 

Camber. — To  give  a  convexity  to  the  upper  surface  of  a  beam. 

Campanile. — A  tower  for  the  reception  of  bells,  usually,  in  Italy,  separated 
from  the  church. 

Canopy. — An  ornamental  covering  over  a  seat  of  state. 

Cantalivers. — The  ends  of  rafters  under  a  projecting  roof.  Pieces  of  wood 
or  stone  supporting  the  eaves. 

Capital. — The  uppermost  part  of  a  column  included  between  the  shaft  and 
the  architrave. 

Caravansera. — In  the  East,  a  large  public  building  for  the  reception  of  trav- 
ellers by  caravans  in  the  desert. 


GLOSSARY.  629 

Carpentry.  —  (From  the  Latin  carpentum,  carved  wood.)  That  department 
of  science  and  art  which  treats  of  the  disposition,  the  construction,  and  the 
relative  strength  of  timber.  The  first  is  called  descriptive,  the  second  con- 
structive, and  the  last  mechanical  carpentry. 

Caryatides. — Figures  of  women  used  instead  of  columns  to  support  an 
entablature. 

Casino. — A  small  country-house. 

Castellated. — Built  with  battlements  and  turrets  in  imitation  of  ancient 
castles. 

Castle. — A  building  fortified  for  military  defence.  A  house  with  towers, 
usually  encompassed  with  walls  and  moats,  and  having  a  donjon,  or  keep,  in 
the  centre. 

Catacombs. — Subterraneous  places  for  burying  the  dead. 

Cathedral. — The  principal  church  of  a  province  or  diocese,  wherein*  the 
throne  of  the  archbishop  or  bishop  is  placed. 

Cavetto. — A  concave  moulding  comprising  the  quadrant  of  a  circle. 

Cemetery. — An  edifice  or  area  where  the  dead  are  interred. 

Cenotaph. — A  monument  erected  to  the  memory  of  a  person  buried  in 
another  place. 

Centring. — The  temporary  woodwork,  or  framing,  whereon  any  vaulted 
work  is  constructed. 

Cesspool. — A  well  under  a  drain  or  pavement  to  receive  the  waste  water  and 
sediment. 

Chamfer. — The  bevelled  edge  of  anything  originally  right  angled. 

Chancel. — That  part  of  a  Gothic  church  in  which  the  altar  is  placed. 

Chantry. — A  little  chapel  in  ancient  churches,  with  an  endowment  for  one 
or  more  priests  to  say  mass  for  the  relief  of  souls  out  of  purgatory. 

Chapel. — A  building  for  religious  worship,  erected  separately  from  a  church, 
and  served  by  a  chaplain. 

Chaplet. — A  moulding  carved  into  beads,  olives,  etc. 

Cincture. — The  ring,  listel,  or  fillet,  at  the  top  and  bottom  of  a  column, 
which  divides  the  shaft  of  the  column  from  its  capital  and  base. 

Circus. — A  straight,  long,  narrow  building  used  by  the  Romans  for  the  ex- 
hibition of  public  spectacles  and  chariot  races.  At  the  present  day,  a  building 
enclosing  an  arena  for  the  exhibition  of  feats  of  horsemanship. 

Clere-story. — The  upper  part  of  the  nave  of  a  church  above  the  roofs  of  the 
aisles. 

Cloister. — The  square  space  attached  to  a  regular  monastery  or  large  church, 
having  a  peristyle  or  ambulatory  around  it,  covered  with  a  range  of  buildings. 

Coffer-dam. — A  case  of  piling,  water-tight,  fixed  in  the  bed  of  a  river,  for  the 
purpose  of  excluding  the  water  while  any  work,  such  as  a  wharf,  wall,  or  the 
pier  of  a  bridge,  is  carried  up. 

Collar-beam. — A  horizontal  beam  framed  between  two  principal  rafters  above 
the  tie-beam. 

Colonnade. — A  range  of  columns. 

Columbarium. — A  pigeon-house. 

Column. — A  vertical  cylindrical  support  under  the  entablature  of  an  order. 

Common-rafters. — The  same  as  jack-rafters,  which  see. 


630  APPENDIX. 

Conduit. — A  long,  narrow,  walled  passage  underground,  for  secret  com- 
munication between  different  apartments.  A  canal  or  pipe  for  the  conveyance 
of  water. 

Conservatory. — A  building  for  preserving  curious  and  rare  exotic  plants. 

Consoles. — The  same  as  ancones,  which  see. 

Contour. — The  external  lines  which  bound  and  terminate  a  figure. 

Convent. — A  building  for  the  reception  of  a  society  of  religious  persons. 

Coping. — Stones  laid  on  the  top  of  a  wall  to  defend  it  from  the  weather. 

Corbels. — Stones  or  timbers  fixed  in  a  wall  to  sustain  the  timbers  of  a  floor 
or  roof. 

Cornice. — Any  moulded  projection  which  crowns  or  finishes  the  part  to 
which  it  is  affixed. 

Corona. — That  part  of  a  cornice  which  is  between  the  crown-moulding  and 
the  bed-mouldings. 

Cornucopia. — The  horn  of  plenty. 

Corridor. — An  open  gallery  or  communication  to  the  different  apartments  of 
a  house. 

Cove. — A  concave  moulding. 

Cripple-rafters. — The  short  rafters  which  are  spiked  to  the  hip-rafter  of  a 
roof. 

Crockets. — In  Gothic  architecture,  the  ornaments  placed  along  the  angles  of 
pediments,  pinnacles,  etc. 

Crosettes. — The  same  as  ancones,  which  see. 

Crypt. — The  under  or  hidden  part  of  a  building. 

Culvert. — An  arched  channel  of  masonry  or  brickwork,  built  beneath  the 
bed  of  a  canal  for  the  purpose  of  conducting  water  under  it.  Any  arched 
channel  for  water  underground. 

Cupola. — A  small  building  on  the  top  of  a  dome. 

Curtail-step. — A  step  with  a  spiral  end,  usually  the  first  of  the  flight. 

Cusps. — The  pendants  of  a  pointed  arch. 

Cyma. — An  ogee.  There  are  two  kinds  ;  the  cyma-recta,  having  the  upper 
part  concave  and  the  lower  convex,  and  the  cyma-reversa,  with  the  upper  part 
convex  and  the  lower  concave. 

Dado. — The  die,  or  part  between  the  base  and  cornice  of  a  pedestal. 

Dairy. — An  apartment  or  building  for  the  preservation  of  milk,  and  the 
manufacture  of  it  into  butter,  cheese,  etc. 

Dead-shoar. — A  piece  of  timber  or  stone  stood  vertically  in  brickwork,  to 
support  a  superincumbent  weight  until  the  brickwork  which  is  to  carry  it 
has  set  or  become  hard. 

Decastyle. — A  building  having  ten  columns  in  front. 

Dentils. — (From  the  Latin,  dentes,  teeth.)  Small  rectangular  blocks  used  in 
the  bed-mouldings  of  some  of  the  orders. 

Diastyle. — An  intercolumniation  of  three,  or,  as  some  say,  four  diameters. 

Die. — That  part  of  a  pedestal  included  between  the  base  and  the  cornice  ;  it 
is  also  called  a  dado. 

Dodecastyle. — A  building  having  twelve  columns  in  front. 

Donjon. — A  massive  tower  within  ancient  castles,  to  which  the  garrison 
might  retreat  in  case  of  necessity. 


GLOSSARY.  631 

Dcoks. — A  Scotch  name  given  to  wooden  brick*. 

Dormer. — A  window  placed  on  the  roof  of  a  house,  the  frame  being  placed 
vertically  on  the  rafters. 

Dormitory. — A  sleeping-room. 

Dovecote. — A  building  for  keeping  tarrje  pigeons.     A  columbarium. 

Echinus. — The  Grecian  ovolo. 

Elevation. — A  geometrical  projection  drawn  on  a  plane  at  right  angles  to 
the  horizon. 

Entablature. — That  part  of  an  order  which  is  supported  by  the  columns  ; 
consisting  of  the  architrave,lrieze,  and  cornice. 

Etistyle.—An  intercolumniation  of  two  and  a  quarter  diameters. 

Exchange. — A  building  in  which  merchants  and  brokers  meet  to  transact 
business. 

Extrados. — The  exterior  curve  of  an  arch. 

Facade. — The  principal  front  of  any  building. 

Face-mould. — The  pattern  for  marking  the  plank  out  of  which  hand-railing 
is  to  be  cut  for  stairs,  etc. 

Facia,  or  Fascia. — A  flat  member,  like  a  band  or  broad  fillet. 

Falling-mould. — The  mould  applied  to  the  convex,  vertical  surface  of  the 
rail-piece,  in  order  to  form  the  back  and  under  surface  of  the  rail,  and  finish 
the  squaring. 

Festoon. — An  ornament  representing  a  wreath  of  flowers  and  leaves. 

Fillet. — A  narrow  flat  band,  listel,  or  annulet,  used  for  the  separation  of 
one  moulding  from  another,  and  to  give  breadth  and  firmness  to  the  edges  of 
mouldings. 

Flutes.— Upright  channels  on  the  shafts  of  columns. 

Flyers. — Steps  in  a  flight  ot  stairs  that  are  parallel  to  each  other. 

Forum. — In  ancient  architecture  a  public  market ;  also,  a  place  where  the 
common  courts  were  held  and  law  pleadings  carried  on. 

Foundry. — A  building  in  which  various  metals  are  cast  into  moulds  or 
shapes. 

Fneze. — That  part  of  an  entablature  included  between  the  architrave  and 
the  corn.ice. 

• 

Gable.— The  vertical,  triangular  piece  of  wall  at  the  end  of  a  roof,  from  the 
level  of  the  eaves  to  the  summit. 

Gain. — A  recess  made  to  receive  a  tenon  or  tusk. 

Gallery.— A  common  passage  to  several  rooms  in  an  upper  story.  A  long 
room  for  the  reception  of  pictures.  A  platform  raised  on  columns,  pilasters, 
or  piers. 

Girder. -rite  principal  beam  in  a  floor,  for  supporting  the  binding  and 
other  joists,  whereby  the  bearing  or  length  is  lessened. 

Glyph— A  vertical,  sunken  channel.  From  their  number,  those  in  the 
Doric  order  are  called  triglyphs. 

Granary.— A  building  for  storing  grain,  especially  that  intended  to  be 
kept  for  a  considerable  time. 


632  APPENDIX. 

Groin. — The  line  formed  by  the  intersection  of  two  arches,  which  cross  each 
other  at  any  angle. 

Gutta. — The  small  cylindrical  pendent  ornaments,  otherwise  called  drops, 
used  in  the  Doric  order  under  the  triglyphs,  and  also  pendent  from  the  mutult 
of  the  cornice. 

Gymnasium. — Originally,  a  place  measured  out  and  covered  with  sand  for 
the  exercise  of  athletic  games  ;  afterward,  spacious  buildings  devoted  to  the 
mental  as  well  as  corporeal  instruction  of  youth. 

Hall. — The  first  large  apartment  on  entering  a  house.  The  public  room  of 
a  corporate  body.  A  manor-house. 

Ham.— A.  house  or  dwelling-place.  A  street  or  village  :  hence  Notting- 
ham,  Buckingham,  etc.  Hamlet,  the  diminutive  of  ham,  is  a  small  street  or 
village. 

Helix. — The  small  volute,  or  twist,  under  the  abacus  in  the  Corinthian 
capital. 

Hem. — The  projecting  spiral  fillet  of  the  Ionic  capital. 

Hexastyle. — A  building  having  six  columns  in  front. 

Hip-rafter. — A  piece  of  timber  placed  at  the  angle  made  by  two  adjacent 
inclined  roofs. 

Homestall. — A  mansion-house,  or  seat  in  the  country. 

Hotel,  or  Hostel. — A  large  inn  or  place  of  public  entertainment.  A  large 
house  or  palace. 

Hot-house. — A  glass  building  used  in  gardening. 

Hovel. — An  open  shed. 

Hut. — A  small  cottage  or  hovel,  generally  constructed  of  earthy  materials, 
as  strong  loamy  clay,  etc. 

Impost. — The  capital  of  a  pier  or  pilaster  which  supports  an  arch. 
Intaglio. — Sculpture  in  which  the  subject  is  hollowed  out,  so  that  the  im- 
pression from  it  presents  the  appearance  of  a  bas-relief. 
Intercolumniation. — The  distance  between  two  columns. 
Intrados. — The  interior  and  lower  curve  of  an  arch. 

Jack-rafters. — Rafters  that  fill  in  between  the  principal  rafters  of  a  roof; 
called  also  common-rafters.  • 

Jail. — A  place  of  legal  confinement. 

Jambs. — The  vertical  sides  of  an  aperture. 

Joggle-piece. — A  post  to  receive  struts. 

Joists.— The  timbers  to  which  the  boards  of  a  floor  or  the  laths  of  a  ceiling 
are  nailed. 

Keep. — The  same  as  donjon,  which  see. 
Key-stone. — The  highest  central  stone  of  an  arch. 

Kiln. — A  building  for  the  accumulation  and  retention  of  heat,  in  order  to 
dry  or  burn  certain  materials  deposited  within  it. 
King-post. — The  centre-post  in  a  trussed  roof. 
Knee. — A  convex  bend  in  the  back  of  a  hand-rail.     See  Ramp. 


GLOSSARY.  633 

Lactarium. — The  same  as  dairy,  which  see. 

Lantern. — A  cupola  having  windows  in  the  sides  for  lighting  an  apartment 
beneath. 

Larmier.— The  same  as  corona,  which  see. 

Lattice. — A  reticulated  window  for  the  admission  of  air,  rather  than  light, 
as  in  dairies  and  cellars. 

Lever-boards. — Blind-slats;  a  set  of  boards  so  fastened  that  they  maybe 
turned  at  any  angle  to  admit  more  or  less  light,  or  to  lap  upon  each  other  so 
as  to  exclude  all  air  or  light  through  apertures. 

Lintel. — A  piece  of  timber  or  stone  placed  horizontally  over  a  door,  win- 
dow, or  other  opening. 

Listel. — The  same  asyf//<?/,  which  see. 

Lobby. — An  enclosed  space,  or  passage,  communicating  with  the  principal 
room  or  rooms  of  a  house. 

Lodge. — A  small  house  near  and  subordinate  to  the  mansion.  A  cottage 
placed  at  the  gate  of  the  road  leading  to  a  mansion. 

Loof.  — A.  small  narrow  window.  Loophole  is  a  term  applied  to  the  vertical 
series  of  doors  in  a  warehouse,  through  which  goods  are  delivered  by  means 
of  a  crane. 

Luffer-boarding. — The  same  as  lever-boards,  which  see. 

Luthetn. — The  same  as  dormer,  which  see. 

Mausoleum. — A  sepulchral  building — so  called  from  a  very  celebrated  one 
erected  to  the  memory  of  Mausolus,  king  of  Caria,  by  his  wife  Artemisia. 

Melopa. — The  square  space  in  the  frieze  between  the  triglyphs  of  the  Doric 
order. 

Mezzanine.— A  story  of  small  height  introduced  between  two  of  greater 
height. 

Minaret. — A  slender,  lofty  turret  having  projecting  balconies,  common  in 
Mohammedan  countries. 

Minster.— A  church  to  which  an  ecclesiastical  fraternity  has  been  or  is 
attached. 

Moat. — An  excavated  reservoir  of  water,  surrounding  a  house,  castle,  or 

town. 

Modillion. — A  projection  under  the  corona  of  the  richer  orders,  resembling 
a  bracket. 

Module.— The  semi-diameter  of  a  column,  used  by  the  architect  as  a  meas- 
ure by  which  to  proportion  the  parts  of  an  order. 

Monastery.— A  building  or  buildings  appropriated  to  the  reception  of 
monks. 

Monopteron. — A  circular  colonnade  supporting  a  dome  without  an  enclos- 
ing wall. 

Mosaic.— A  mode  of  representing  objects  by  the  inlaying  of  small  cubes  of 
glass,  stone,  marble,  shells,  etc. 

Mosque. — A  Mohammedan  temple  or  place  of  worship. 

Midlions.—Thz  upright  posts  or  bars  which  divide  the  lights  in  a  Gothic 
window. 

Mtiniment-house.—k  strong,  fire-proof  apartment  for  the  keeping  and  pres- 
ervation of  evidences,  charters,  seals,  etc.,  called  muniments. 


634  APPENDIX. 

Museum. — A  repository  of  natural,  scientific,  and  literary  curiosities  or  of 
works  of  art. 

Mutule. — A  projecting  ornament  of  the  Doric  cornice  supposed  to  repre- 
sent the  ends  of  rafters. 

Nave. — The  main  body  of  a  Gothic  church. 
Newel. — A  post  at  the  starting  or  landing  of  a  flight  of  stairs. 
Niche. — A  cavity  or  hollow  place  in  a  wall  for  the  reception  of  a  statue, 
vase,  etc. 

Nogs. — Wooden  bricks. 

Nosing. — The  rounded  and  projecting  edge  of  a  step  in  stairs. 

Nunnery. — A  building  or  buildings  appropriated  for  the  reception  of  nuns. 

Obelisk. — A  lofty  pillar  of  a  rectangular  form. 

Octastyle. — A  building  with  eight  columns  in  front. 

Odeum. — Among  the  Greeks,  a  species  of  theatre  wherein  the  poets  and 
musicians  rehearsed  their  compositions  previous  to  the  public  production  of 
them. 

Ogee. — See  cyma. 

Orangery. — A  gallery  or  building  in  a  garden  or  parterre  fronting  the 
south. 

Oriel-window. — A  large  bay  or  recessed  window  in  a  hall,  chapel,  or  other 
apartment. 

Ovolo. — A  convex  projecting  moulding  whose  profile  is  the  quadrant  of  a 
circle. 

Pagoda. — A  temple  or  place  of  worship  in  India. 

Palisade. — A  fence  of  pales  or  stakes  driven  into  the  ground. 

Parapet. — A  small  wall  of  any  material  for  protection  on  the  sides  of 
bridges,  quays,  or  high  buildings. 

Pavilion. — A  turret  or  small  building  generally  insulated  and  comprised 
under  a  single  roof. 

Pedestal. — A  square  foundation  used  to  elevate  and  sustain  a  column, 
statue,  etc. 

Pediment. — The  triangular  crowning  part  of  a  portico  or  aperture  which 
terminates  vertically  the  sloping  parts  of  the  roof;  this,  in  Gothic  architecture, 
is  called  a  gable. 

Penitentiary. — A  prison  for  the  confinement  of  criminals  whose  crimes  are 
not  of  a  very  heinous  nature. 

Piazza. — A  square,  open  space  surrounded  by  buildings.  This  term  is 
often  improperly  used  to  denote  a  portico. 

Pier. — A  rectangular  pillar  without  any  regular  base  or  capital.  The  up- 
right, narrow  portions  of  walls  between  doors  and  windows  are  known  by  this 
term. 

Pilaster. — A  square  pillar,  sometimes  insulated,  but  more  commonly  en- 
gaged in  a  wall,  and  projecting  only  a  part  of  its  thickness. 

Piles. — Large  timbers  driven  into  the  ground  to  make  a  secure  foundation 
in  marshy  places,  or  in  the  bed  of  a  river. 


GLOSSARY.  635 

Pitta*.— A  column  of  irregular  form,  always  disengaged,  and  always  deviat- 
ing from  the  proportions  of  the  orders;  whence  the  distinction  between  a 
pillar  and  a  column. 

Pinnacle. — A  small  spire  used  to  ornament  Gothic  buildings. 

Planceer. — The  same  as  soffit,  which  see. 

Plinth. — The  lower  square  member  of  the  base  of  a  column,  pedestal  or 
wall. 

Porch. — An  exterior  appendage  to  a  building,  forming  a  covered  approach 
to  one  of  its  principal  doorways. 

Portal.—  The  arch  over  a  door  or  gate  ;  the  framework  of  the  gate  ;  the 
lesser  gate,  when  there  are  two  of  different  dimensions  at  one  entrance. 

Portcullis.— A  strong  timber  gate  to  old  castles,  made  to  slide  up  and 
down  vertically. 

Portico. — A  colonnade  supporting  a  shelter  over  a  walk,  or  ambulatory. 

Priory. — A  building  similar  in  its  constitution  to  a  monastery  or  abbey, 
the  head  whereof  was  called  a  prior  or  prioress. 

Prism. — A  solid  bounded  on  the  sides  by  parallelograms,  and  on  the  ends 
by  polygonal  figures  in  parallel  planes. 

Prostyle. — A  building  with  columns  in  front  only. 

Purlines. — Those  pieces  of  timber  which  lie  under  and  at  right  angles  to 
the  rafters  to  prevent  them  from  sinking. 

Pycnostyle. — An  intercolumniation  of  one  and  a  half  diameters. 

Pyramid. — A  solid  body  standing  on  a  square,  triangle,  or  potygonal  basis 
and  terminating  in  a  point  at  the  top. 

Quarry. — A  place  whence  stones  and  slates  are  procured. 
Quay. — (Pronounced  key.)    A  bank  formed  towards  the  sea  or  on  the  side 
of  a  river  for  free  passage,  or  for  the  purpose  of  unloading  merchandise. 
Quoin. — An  external  angle.     See  Rustic  quoins. 

Rabbet,  or  Rebate. — A  groove  or  channel  in  the  edge  of  a  board. 

Ramp. — A  concave  bend  in  the  back  of  a  hand-rail. 

Rampant  arch. — One  having  abutments  of  different  heights.    . 

Regula. — The  band  below  the  taenia  in  the  Doric  order. 

Riser. — In  stairs,  the  vertical  board  forming  the  front  of  a  step. 

Rostrum. — An  elevated  platform  from  which  a  speaker  addresses  an  audi- 
ence. 

Rotunda. — A  circular  building. 

Rubble-wall. — A  wall  built  of  unhewn  stone. 

Rudenture. — The  same  as  cable,  which  see. 

Rustic  quoins. — The  stones  placed  on  the  external  angle  of  a  building,  pro- 
jecting beyond  the  face  of  the  wall,  and  having  their  edges  bevelled. 

Rustic-work. — A  mode  of  building  masonry  wherein  the  faces  of  the  stones 
are  left  rough,  the  sides  only  being  wrought  smooth  where  the  union  of  the 
stones  takes  place. 

Salon,  or  Saloon. — A  lofty  and  spacious  apartment  comprehending  the 
height  of  two  stories  with  two  tiers  of  windows. 


636  APPENDIX. 

Sarcophagus. — A  tomb  or  coffin  made  of  one  stone. 

Scantling. — The  measure  to  which  a  piece  of  timber  is  to  be  or  has  been 
cut. 

Scarfing. — The  joining  of  two  pieces  of  timber  by  bolting  or  nailing  trans- 
versely together,  so  that  the  two  appear  but  one. 

Scotia. — The  hollow  moulding  in  the  base  of  a  column,  between  the  fillets 
of  the  tori. 

Scroll. — A  carved  curvilinear  ornament,  somewhat  resembling  in  profile 
the  turnings  of  a  ram's  horn. 

Sepulchre. — A  grave,  tomb,  or  place  of  interment. 

Sewer. — A  drain  or  conduit  for  carrying  off  soil  or  water  from  any  place. 

Shaft. — The  cylindrical  part  between  the  base  and  the  capital  of  a  column. 

Shoar. — A  piece  of  timber  placed  in  an  oblique  direction  to  support  a 
building  or  wall. 

Sill. — The  horizontal  piece  of  timber  at  the  bottom  of  framing  ;  the  timber 
or  stone  at  the  bottom  of  doors  and  windows. 

Soffit. — The  underside  of  an  architrave,  corona,  etc.  The  underside  of 
the  heads  of  doors,  windows,  etc. 

Summer. — The  lintel  of  a  door  of  window  ;  a  beam  tenoned  into  a  girder 
to  support  the  ends  of  joists  on  both  sides  of  it. 

Systyle.- — An  intercolumniation  of  two  diameters. 

Tania. — The  fillet  which  separates  the  Doric  frieze  from  the  architrave. 

Talus. — The  slope  or  inclination  of  a  wall,  among  workmen  called  bat- 
tering. 

Terrace. — An  area  raised  before  a  building,  above  the  level  of  the  ground, 
to  serve  as  a  walk. 

Tesselated  pavement. — A  curious  pavement  of  mosaic  work,  composed  of 
small  square  stones. 

Tetrastyle. — A  building  having  four  columns  in  front. 

Thatch. — A  covering  of  straw  or  reeds  used  on  the  roofs  of  cottages, 
barns,  etc. 

Theatre. — A  building  appropriated  to  the  representation  of  dramatic 
spectacles. 

Tile. — A  thin  piece  or  plate  of  baked  clay  or  other  material  used  for  the 
external  covering  of  a  roof. 

Tomb. — A  grave,  or  place  for  the  interment  of  a  human  body,  including 
also  any  commemorative  monument  raised  over  such  a  place. 

Torus. — A  moulding  of  semi-circular  profile  used  in  the  bases  of  col- 
umns. 

Tower. — A  lofty  building  of  several  stories,  round  or  polygonal. 

Transept. — The  transverse  portion  of  a  cruciform  church. 

Transom. — The  b^am  across  a  double-lighted  window  ;  if  the  window 
have  no  transom,  it  is  called  a  clere-story  window. 

Thread. — That  part  of  a  step  which  is  included  between  the  face  of  its  riser 
and  that  of  the  riser  above. 

Trellis. — A  reticulated  framing  made  of  thin  bars  of  wood  for  screens,  win- 
dows, etc. 


GLOSSARY.  637 

\ 

Tiiglyph.—i:\\e.  vertical  tablets  in  the  Doric  frieze,  chamfered  on  the  two 
vertical  edges,  and  having  two  channels  in  the  middle. 

Tripod. — A  table  or  seat  with  three  legs. 

Trochilus. — The  same  as  scotia,  which  see. 

Truss. — An  arrangement  of  timbers  for  increasing  the  resistance  to  cross- 
strains,  consisting  of  a  tie,  two  struts,  and  a  suspending-piece. 

Turret. — A  small  tower,  often  crowning  the  angle  of  a  wall,  etc. 

Tusk. — A  short  projection  under  a  tenon  to  increase  its  strength. 

Tympanum. — The  naked  face  of  a  pediment,  included  between  the  level  and 
the  raking  mouldings. 

Underpinning. — The  wall  under  the  ground-sills  of  a  building. 

University. — An  assemblage  of  colleges  under  the  supervision  of  a  senate,  etc. 

Vault. — A  concave  arched  ceiling  resting  upon  two  opposite  parallel  walls. 

Venetian-door. — A  door  having  side-lights. 

Venetian-window. — A  window  having  three  separate  apertures. 

Veranda. — An  awning.  An  open  portico  under  the  extended  roof  of  a 
building. 

Vestibule. — An  apartment  which  serves  as  a  medium  of  communication  to 
another  room  or  series  of  rooms. 

Vestry. — An  apartment  in  a  church,  or  attached  to  it,  for  the  preservation 
of  the  sacred  vestments  and  utensils. 

Villa. — A  country-house  for  the  residence  of  an  opulent  person. 

Vinery. — A  house  for  the  cultivation  of  vines. 

Volute. — A  spiral  scroll,  which  forms  the  principal  feature  of  the  Ionic  and 
the  Composite  capitals. 

Vottssoirs. — Arch-stones. 

Wainscoting. — Wooden  lining  of  walls,  generally  in  panels. 

Water-table. — The  stone  covering  to  the  projecting  foundation  or  other  walls 
of  a  building. 

Well. — The  space  occupied  by  a  flight  of  stairs.  The  space  left  beyond  the 
ends  of  the  steps  is  called  the  well-hole. 

Wicket. — A  small  door  made  in  a  gate. 

Winders. — In  stairs,  steps  not  parallel  to  each  other. 

Zophorus. — The  same  as  frieze,  which  see. 

Zystos. — Among  the  ancients,  a  portico  of  unusual  length,  commonly  appro- 
priated  to  gymnastic  exercises. 


638  APPENDIX. 

TABLE  OF  SQUARES,  CUBES.  AND  ROOTS. 

(From  Button's  Mathematics.) 


No. 

Square. 

Cube. 

Sq.  Root.  jCubeRooJ|  No. 

Square. 

Cube. 

Sq.  Root. 

CubeR-wt. 

rr 

1 

1 

1-0000000  1-000000 

68 

4624 

314432 

8-2462113 

4-081655 

2 

4 

8 

1  4142136 

1-259921 

69 

4761 

328509 

8-3066239 

4  1015G6 

3 

9 

27 

1-7320508 

1-442250 

70 

4900 

343000 

8-3666003  4-121285 

4 

16 

64 

2-0000000 

1-537401 

71 

5041 

357911 

8-4261498!  4-140818 

5 

25 

125 

2-2360680 

1-709976 

72 

5184 

373248 

8-4852814J  4-160168 

6 

36 

216 

2-4494897 

1-817121 

73 

5329 

389017 

8-54400371  4-179339 

7 

49 

343 

2-6457513 

1-912931 

74 

5476 

405224 

8-6023253]  4-198336 

8 

64 

512 

2-8284271 

2-000000 

75 

5625 

421875 

8-6602540  4  217163 

9 

81 

729 

3-0000000 

2-080034 

76 

5776 

433976 

8-7177979  4235324 

10 

100 

1000 

3-1622777 

2-154435 

77 

5929 

456533 

8-7749644 

4-254321 

11 

121 

1331 

3-3166243 

2-223330 

78 

6084 

474552 

8-8317609!  4-272659 

12 

144 

1728 

3-4641016 

2-239429 

79 

6241 

493039 

8-8881944  4-290840 

13 

169 

2197 

36055513  2351335 

80 

6400 

512000 

8-9442719!  4-30SS69 

14 

196 

2744 

3-74165741  2-410142 

81 

6561 

531441 

9-(M)0()000|  4-3*6749 

15 

225 

3375 

38729833!  2-466212 

82 

6724 

551358 

9-05538511  4-344431 

16 

256 

4096 

4'OOOOOOOi  2-519842; 

83 

6339 

571787 

9-1104336'  4-362071 

17 

239 

4913 

4-12310561  2-571232 

84 

7056 

592704 

9-1651514!  4-379519 

18 

324 

5332 

4-2426407 

2-62074  1  ' 

85 

7225 

614125 

9-2195445  4-396830 

19 

361 

635J 

4-3533989  2-663402! 

86 

7396 

636055 

9-2736185  4-414005 

20 

400 

8000 

4-4721350  2-714118 

87 

7569 

653503 

9-3273791  4-431048 

21 

441 

9261 

4-5825757 

2-758924 

88 

7744 

681472 

9-38J8315J  4-447960 

22 

484 

10643 

4-6904158  2-802033 

89 

7921 

704969 

9-4339311!  4  "464745 

23 

529 

12167 

4-79533151  2-843367 

90 

8100 

729000 

9-4363330  4  -58  1405 

24 

576 

13324 

4-8989795]  2-881499 

91 

8281 

753571 

9-5393920]  4-497941 

25 

625 

15625 

5-0000000!  2-924018 

92 

8464 

778688 

9-5916630  4514357 

26 

676 

17576 

5  -09901  95  j  2-962496 

93 

8649 

804357 

9-64365081  4-530055 

27 

729 

19633 

5-1961524'  3-000000 

94 

8836 

830534 

9-6953597 

4  516336 

28 

784 

21952 

529151)26  3-036539 

95 

9025 

857375 

9-7467943 

4-5o2903 

29 

841 

24389 

5-3851648  3-072317 

96 

9216 

884736 

9-7979590 

4-573357 

30 

900 

27000 

5-4772256  3-107232 

97 

9409 

912673 

9-8483578 

4-591701 

31 

961 

29791 

556776441  3-14133l| 

98 

9604 

941192 

9-8994949 

4-610436 

32 

1024 

32763 

5-6568542  3-1748021 

99 

9801 

970299 

9-9493744 

4-626065 

33 

1089 

35937 

57445626  3-20753i!  100 

10000 

1000000 

10-0000000 

4641589 

34 

1156 

39304 

5-83J9519  3233612  101 

10201 

1030301 

10-0498756 

4-657009 

35 

1225 

42875 

5-9160798  3-271066 

102 

10404 

1061208 

10-0995049  41372329 

36 

1296 

46656 

6-0000000 

3-331927 

103 

10609 

1092727 

10-  14839  1GJ  4-687548 

37 

1369 

50653 

6-0327625 

3-332222 

104 

10816 

1124864 

10-1980330!  4-702659 

1  38 

1444 

54872 

6-1644140 

3-361975 

105 

11025 

1157625 

10-2469503  4-717694 

«9 

1521 

59319 

6-2449980 

3-331211 

106 

11236 

1191016 

10-2956301!  4-732623 

40 

1600 

6401)0 

6-3245553 

3419952 

107 

11449 

1225043 

10-3440801]  4-747459 

41 

1681 

68921 

6-4031242  3448217 

108 

11664 

1259712 

10-3923343'  4-762203 

\J 

1764 

74088 

6-4307407  3-476027 

109 

11831 

1295029 

10-4403J65'  4-776356 

^3 

1849 

79507 

6-5574335 

3503398 

110 

12100 

1331000 

10-4880335  4-791420 

44 

1936 

85184 

6-6332496 

3530348 

111 

12321 

1367631 

10-535G53S!  4-805895 

45 

2025 

91125 

6-7082039 

3-556893 

112 

12544 

1404928 

10-58300521  4-820284 

46 

2116 

97336 

6-7823300 

3-533J48 

113 

12769 

1442897 

10-6301453  4-834533 

47 

2209 

103423 

6-8556546  3-608325 

114 

12996 

1481544 

10-0770783 

4-848808 

48 

2304 

1  10592 

69232032  3634241 

115 

13225 

1520875 

107238053 

4-862944 

49 

2401 

117649 

7-0000000;  3-6593J6 

116 

13456 

1560896 

10-7703296 

4-87G999 

50 

2500 

125000 

7-0710678  3634031 

117 

13689 

1601613 

10-816553$ 

4-890973 

51 

2601 

132651 

7-1414284  3-70843J 

118 

13924 

1643032 

10-8627805 

4-904863 

52 

2704 

140608 

7-2111026;  3-732511 

119 

14161 

1685159 

10-9087121, 

4-913685 

53 

2809 

148877 

7-2301099  3-756236 

120 

14400 

1728000 

10-9544512 

4-93-3424 

54 

2916 

157464 

7-3181692  3-779763  121 

14641 

1771561 

11-0000000 

4-946087 

55 

3025 

166375 

7-4161985!  3-502952  !  122 

14884 

1815848 

11-0453610 

4-959676 

56 

.3136 

175616 

7-4833148 

3-8258521!  123 

15129 

1860867 

11*0905365 

4-973190 

57 

3219 

185193 

7-549334  1 

3-343501  1!  124 

15376 

1906624 

11-1355287 

4-986631 

58 

3364 

195112 

7-6157731 

3-870877  125 

15625 

1953125 

11-1803399 

5-000000 

59 

3481 

205379 

7-6311457 

3-892996  j  126 

15876 

2000376 

11-2219722 

5-013293 

60 

3600 

216001) 

7-7459667 

3-914868  127 

16129 

2048383 

11-2694277 

5-026526 

61 

3721 

226981 

7-8102497 

3-936497 

128 

16334 

2097152 

11-3137085 

5-039684 

62 

3344 

233328 

7-8740079 

3-957891 

129 

16641 

2146689 

1  1-35  Vd  167 

5-052774 

63 

3969 

250017 

7-9372539 

3-979057 

130 

16900 

2197000  114017543 

5-065797 

64 

4096 

262144 

8-0000000 

4-000000 

131 

17161 

2248091  11*4455331 

5-078753 

65 

4225 

274625 

8-0622577 

4-020726  132 

17424 

2299968  11-4891253 

5-091643 

66 

4356 

2374% 

8-1240334 

4-041240  133 

17689 

2252637  11-53256261  5104469 

67 

448'J 

300763 

8-1853523  4-061548|!  134 

17956   2406I041  ]i-575«369l  5'  11  7230 

TABLE   OF  SQUARES,   CUBES,   AND   ROOTS. 


639 


tfo. 

Square. 

Cube. 

Sq.  Root.  CubeRoot. 

No. 

Square. 

40804 
41209 

Cube.    Sq.  Root.  CubcRooL 

135 
136 

18225 
18496 

2460375 
2515156 

11-6189500  5-129928 
11-6619033!  5-142563 

202 
203 

!   8242406 
8365427 

14-2126704  5-867464 
14-24780681  5-877131 

137 

18769 

2571353 

11-7016999J  5-155137 

204 

4I616J   8489664 

14-23285691  5-836765 

133 

19044 

2628072 

11-7473401 

5-  167649 

205 

42023 

8615125!  14-3178211 

5-896368 

139 

19321 

2635619 

11-7898261 

5-180101 

206  42  1ST 

8741816!  14-3527001 

5-905941 

140 

19600 

27-14000 

11-83215% 

5-192494 

207 

42849 

8369743  14'337494fi 

5-915432 

141 

19881 

2803221 

11-8743422 

5-204328 

203 

43264 

8998912  11-4222051 

5-924992 

142 

20164 

2S63283 

11-9163753 

5-217103 

209 

43681 

9129329  14-4568322 

5-934473 

143 

20449 

2924207 

11-9582607 

5-229321 

210 

44100 

9261000  14-4913767 

5-943J22 

144 

20736 

2985934 

12-0000000 

5-241483 

211 

44521 

9393931]  14-5253390 

5-953342 

145 

21025 

3048625 

12-0415946 

5-253533 

212 

44944 

9528123;  14-5602193!  5-962732 

146 

21316 

3112136 

12-0830460J  5-265637 

213 

45369 

9663597J  14-5945195!  5-972093 

147 

21609 

317C>523 

12-1243557  5-277632 

214 

45796 

9800344!  146287338!  5-931424 

148 

21904 

3241792 

12-1655251 

5-289572! 

215 

46225 

99333751  14-6623783!  S'%0726 

149 

22201 

3307949 

12-2065556 

5301459! 

216 

46656 

10077696 

14-69693851  6-000000 

150 

22500 

3375000 

12-2174487 

5-3132931 

217 

47089 

10218313 

14-73091991  6-009245 

1511  22301 

3142951 

12-2332057 

5-325074 

218 

47524 

10.360232!  14-7648231 

6-018462 

152i  23104 

3511808 

12-3238280 

5-336803^ 

219 

47961 

10503459!  14-7986486 

6-027650 

153  23409 

3531577 

12-3693169 

5-348481! 

220 

48400  10648000  14-8323970 

6-036811 

154  23710 

3652264 

12-4096736 

5-360108! 

221 

48341,  10793861  14-8660687 

6-045943 

155  24025   3723375 

12-449899.  » 

5-371685 

222 

49234   10941048!  14-89  J6044 

6-055049 

156  24336 

37%  416i  12-4399960 

5-383213 

223 

49729 

11039567  14-9331845 

6-064127 

157  24649   3869393 

12-5299641 

5-394691 

224 

50176 

11239424  149666295  6-073178 

153  24964 

3944312 

12-5698051 

5-406120J 

225 

50625 

11390625  15-000000ol  6-082202 

159  25281 

4019679 

12-61)95202 

5-417501J 

226 

51076 

11543176  15-0332964  6-091199 

160J  25600 

4096000 

12-6191106 

5-428335! 

227 

51529 

11697083  15-05651921  6-100170 

161 

25921 

4173281 

12-6385775 

5-440122 

223 

51984 

11852352  15-0996639J  6109115 

162 

26344 

4251523 

12-7279221 

5-451362 

229 

52441 

12008939  15-1327460'  6-118033 

163  26569 

4330747 

12-7671453 

5-462556 

230 

52900 

12167000  15-1657509 

6-126926 

164 

26896 

4410944 

12-8062485 

5-473704  231 

53361 

12326391 

151936342 

6-135792 

165 

27225 

4492125 

12-8452326 

5-484807  232 

53824 

12487168  15-2315462 

6-144634 

166 

87556 

4574296!  12-8340987 

5-495365  233 

54289 

12649337  152643375 

6-153449 

167  27839 

4657463  129228480 

5-506878! 

234 

54756 

12812904 

15-2970585 

6-162240 

168  23^4 

4741632  12-9614814 

5-51781H  -235 

55225 

12977875 

15-32J70J7 

6-171006 

169  28561 

4826309  13-0000000 

5-528775!  236 

55696 

13144256  153622915  6-179747 

170  28'JOO 

4913000  13-0331048 

5-539658  237 

56169 

13312053!  15-3J43043I  6-183463 

171 

2;>2  1  1 

5000211  13-0766968 

5-550499  233 

56644 

134812721  154272486  6-197154 

172 

29581 

5083148  13-1143770 

5-561293!  239 

57121 

13651919!  15-4596248  6-205822 

173  89929]   5177717 

1741  30276   52681)24 

13-1529464 

13-1909060 

5-572055 
5  532770 

240 
241 

57600 
53031 

13324000  15-4919334!  6-214465 
139/7521:  155241747  6223084 

175 

30625 

5359375 

13-2287565 

5-593445' 

242 

58564 

11172433!  15-5563492  6231630 

176 

30976 

5151776  13-2664992  5-604079! 

243 

59049 

14348907 

l.V.'.ssif,:;}  J6-210J51 

177 

31329   5515233  13-3041347 

31C81   5639752  13-3416541 

5-614672! 
5-625226 

244 
245 

59536 

60025 

14526784 
14706125 

15-6204994  6243300 
15-6524753  6-25  7325 

179 

32041   5735339!  13-3790332 

5-635741 

246 

60516 

14836936 

15-6343371  6-265327 

130 

32103   5832000 

13-4161071) 

5-6462  16 

247 

61009 

15069223 

15-7162335;  6-274305 

181!  32761]   5921)711 

13-4536240  5-656653||  248 

61504 

152529J2 

15-74801571  6--N2701 

182i  33124   6023568 

134907376  5-6670511  249 

62001 

15433249 

15-7797333  6-291195 

183;  33  kd   6128487 

13-5277493!  5-67741  1|  250 

62500 

15525000 

15-8113333!  6-299605 

181   33356 

6229504 

13-5646600!  5-6S7734  251 

63001 

15313251 

15-8429795  6-307994 

185   3  1225 

6331625 

13-6014705J  5-693019 

252 

63504 

16003008 

158745079  6316360 

18* 

34596 

613H56 

13-63318171  5-70821'.;  253 

64009 

16194277 

•-•737 

6-321704 

187 

3iy;>'.i 

653921)3 

13-07  17J  13  5-718  17'J  -J.M 

64516 

16337064 

15-9373775 

6-333026 

IMS   35314 

6644672 

13-7113092  5-728.-,:,l 

255  65025 

16581375 

159687194 

6341326 

isj  35721 

6751269 

13-7477271  5-7337J4 

256  65536 

16777216  160000000 

6-349604 

190J  36100 

6859000 

13-78404831  5-1  18391 

257  65049 

16.)74593 

16-0312195 

6-357861 

191   36181 
192  36864 

6967871 
7077838 

13-S202750  5-758966 
13-85640651  5-76-. 

253  66564 
259  67031 

17173312  16-06237S1 
17373979  16-0934769 

6-366097 

6374311 

193i  372)9 

7189057 

13-8924140!  5  -77S.»96  200'  67600 

l?f>76000 

16-1215155 

6-382504 

194i  37,136 

73.H331 

13-9283383;  5-783960 

261  63121 

17779.-.S1  1C,-  1.  V.I.I  11 

6-390676 

195   3302.", 

7414875 

13-9642400J  5-798390 

2621  6S644 

17984723  16-1884141 

6-398889 

196J  38416 

7529536 

14-0000000!  5-808786 

263 

69169 

18191447  16-2172717 

6-406953 

197j  33909 

7615373 

14-0356683  5-818648 

264 

69696 

1SW9711  l:-.2H07t-,< 

6-415069 

198  392u4 

77IWW2 

14-0712473  5-S2-U77 

265 

70225 

18609625  16-2738206 

(•,•12315- 

199)  39H01 

7880599 

14-  10673(50 

5-833272 

265 

70756   188210961  16-30.).",.  >C,  I  6-1312;!- 

2«XJ'  10000 

8000000 

14-1421356 

5-848035 

267 

7I-2S9  190341631  16-3401346  6-439277 

201   40101 

8120601 

14-1774469 

5-857766 

268  71824 

19248832,  16  -37070."):)  6-447306 

640 


APPENDIX. 


No.  Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

No.  Square. 

Cube. 

Sq  Root.  IcubeRnot- 

269'  72361 

19465109 

16-40121951  6-455315 

336 

1  12896 

37933056 

18-3303028 

6952053 

270 

72900 

19633000 

16-43167671  6-463304 

337 

113569 

38272753 

183575598 

6-958943 

271 

73441 

19902511 

16-4620776  6-471274 

333 

114244 

33614472 

18-3347763 

6-965820 

272 

73384 

20123648 

16-4924225!  6-479224 

339 

114921 

38908219 

18-4119526 

6-972683 

273 

74529 

20346417 

16-5227116 

6-487154 

340 

115600 

393(UOOO 

18-4390889 

6-979532 

274 

75076 

20570824 

16-5529454 

6-495065 

341 

116281 

39651821 

184661853 

6-986368 

275 

75625 

20796375 

16-5331240 

6-502357 

342 

116964 

40001638 

18-4932420 

6-993191 

276 

76176 

21024576 

16-6132477 

6-510833 

343 

117649 

40353607 

18-5202592 

7-000000 

277 

7^729 

21253933 

16-6433170 

6-518634 

344 

1  18336 

40707584 

18-5472370 

7-006796 

278 

77234 

21484952 

16-6733320 

6-526519 

345 

119025 

41063625 

18-5741756 

7-013579 

279 

77811 

21717639 

16-7032931 

6534335 

346 

119716 

41421736 

18-6010752 

7-020349 

230 

78400 

21952000 

16-7332005 

6-542133 

347 

120409 

41781923 

18-6279360 

7027106 

281 

78961 

22188041 

16-7630546 

6-549912 

343 

121104 

42144192 

18-6547581 

7-033850 

282 

79524 

22425763 

16-7923556 

6-557672 

349 

121801 

42508549 

18-8815417 

7-040581 

233 

80089 

22665187 

16-8226033 

6565414 

350 

122500 

42875000 

18-7032869 

7-047299 

284 

80656 

22906334 

16-8522995 

6-573139 

351 

123201 

43243551 

18-7349940 

7-054004 

235 

81225 

23149125 

16-8819430 

6-530344 

352 

123904 

43614208 

18-7616630 

7-060697 

286 

81796 

23393656 

169115345 

6538532 

353 

124609 

43985977 

18-7882942 

7-067377 

287 

82369 

23639903 

16-9410743 

6-596202 

354 

125316 

44361864 

18-8148877 

7-074044 

283 

82944 

23387872 

16-9705627 

6-603354 

355 

126025 

44738875 

18-8414437 

7-080699 

239 

83^.21 

24137569 

17-0000000 

6-611489 

356 

126736 

45118016 

18-8679623 

7-087341 

290 

84100,  24389000 

17-0293864  6-619106 

357 

127449 

45499293 

18-8944436 

7-093971 

291 

84681  24642171 

17-0537221  6-626705 

358 

128164 

45882712 

18-9208379 

7-100588 

292 

85264  24897083 

17-0380075  6634237 

359 

128881 

46268279 

18-9472953 

7-107194 

293  85849  251^3757  17-1172428 

6641852 

360 

129600 

46656000 

18-9736660 

7-113787 

294  86  136  25*12184!  17-1464232 

6-649400 

361 

130321 

47045381 

19-0000000 

7-120367 

295  87025 

25672375 

17-1755640 

6-656930 

352 

131044 

47437928 

19-0262976 

7-126936 

296  87616 

25934336 

17-2046505 

6-664444 

363 

131769 

47832147 

19-0525589 

7-133492 

297.  83209 

26198073 

17-2336879 

6-671940 

354 

132496 

48228544 

19-0787840 

7-140037 

293  83804 

26463592 

17-2626765 

6-679420 

365 

133225 

48627125 

19-1049732 

7-146569 

299!  89401 

26730899 

17-2916165 

6-686833 

366 

133956 

49027396 

19-1311265 

7153090 

300  90000 

27000000 

17-3205081 

6-694329 

367 

134689 

49430863 

19-1572441 

7-159599 

301 

90601 

27270901 

17-3493516 

6-701759 

368 

135424 

49835032 

1'.)-  1833261 

7-166096 

302 

91204 

27543603 

17-3781472 

6-709173 

369 

136161 

53243409 

192093727 

7-172531 

303 

91809  27818127 

17-4068952 

6-716570 

370 

136900 

50653000 

19-2353341 

7-179054 

304 

92416 

28094464J  17-4355953 

6-723951 

371 

137641 

51064811 

19-2613603 

7-185516 

305 

93025 

28372625  17-4642492 

6-731316 

372 

138384 

51478848 

19-2373015 

7-191966 

306 

93636 

23652616  17-4928557!  6-733664 

373 

139129 

51895117 

19-3132079 

7-198405 

307 

94249 

28934443.  17-5214155 

6-745997 

374 

139876 

52313624 

19-3390796 

7-204832 

308 

94864 

29218112  17-5499288 

6-753313 

375 

140625 

52734375 

19-3649167 

7-211248 

309 

95481 

2951)3629  17-5783953 

6-760614 

376 

141376 

53157376 

19-3307194 

7-217652 

310 

96100 

297910001  17-6068169 

6-767899 

377 

142129 

53582633 

19-4164878 

7-224045 

311 

96721 

30080231 

17-6351921 

6-775169 

378 

142884 

54010152 

19-4422221 

7-233427 

312 

97344 

33371328 

17-6635217 

6-782423 

379 

143541 

54439939 

19-4679223 

7-236797 

313 

97969;  30664297 

17-6918060 

6-789661 

380 

144400 

54872000 

19-4935837 

7-243156 

311 

985961  30959144 

17-7200451 

6-796834 

331 

145161 

55306341 

19-5192213 

7-249504 

315 

99225 

31255375 

17-7482393 

6-804092 

332 

145924 

55742968 

19-5448203 

7-255841 

31li 

99856 

31554496  17-7763383 

6-811235 

333 

146639 

56181837 

19-5703353 

7-262167 

317 

100489 

31855013  17-8044933 

6-818462 

334 

147456 

56623104 

19-5959179 

7-268482 

318  101124 

32157432  17-8325545 

6-825624 

335 

148225 

57066625 

19-6214169 

7-274786 

3191  101761 

32461759  17-8605711 

6-832771 

336 

148996 

57512456 

19-6468327 

7-231079 

320!  102400 

32763000  17-8835438 

6-839904 

3S7 

149769 

57960603 

19-6723156 

7287362 

321|  103041 

33076161  17-9164729 

6847021 

338 

150544 

58411072 

19-6977  15f 

7-293633 

322!  103684 

33336248i  17-9443534 

6854124 

339 

151321 

53863869  19-7230829 

7-299894 

323  104329 

33598267  17-9722308 

6-861212 

390 

152100 

59319000  19-7434177 

7-306144 

324  104976 

34012224  18-0000000 

6-868235 

391 

152831 

59776471  19-7737199 

7312333 

325  105625 

34323125  18-0277564 

6-875344 

392 

153664 

60236238  19-7989899 

7-318611 

326 

106276 

34645976 

18-0554701 

6-882339 

393 

154449 

60693457  19-8242276 

7-324829 

327 

106929 

34965783  18-0831413 

6-889419 

394 

155236  61162984J  19-8494332 

7331037 

328 

107584 

35287552 

18-1107703 

6-896435 

395 

156025  61623875 

19-8746069 

7-337234 

329 

108241 

35611239 

18-1333571 

6903436 

396 

155816 

62099136 

19-8997487 

7-343420 

33C 

108900 

35937000 

18-1659021 

6-910423 

1  397 

157609 

62570773 

19-9243538 

7-349597 

331 

109561 

36264691  18-1934054 

6-917396 

!  398 

158404 

63044792 

19-9499373 

7-355762 

332  110224 

365943681  18-2208672 

6-924356 

399 

159201 

63521199 

19-9749844 

7361918 

333  110832 

36926037  18-248237G 

6-93130 

400 

160000 

64000000 

20-0030001 

7-363063 

331 

11155fi 

37259704  18-2756669 

6-933232 

401 

160801 

644H1201 

200249844 

7-374198 

335 

112225 

37595375  18-3030052 

6-945150 

!  402 

161604!  64964308 

20  049937* 

7-330323 

TABLE  OF  SQUARES,   CUBES,  AND   ROOTS. 


641 


«0. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

No.  Square. 

Cube.     Sq.  Root. 

CubeRoot. 

403 

1024091  65450327 

20-0748599 

7336437 

470!  22J900!  10332300o|  21-6794334 

7-774930 

404 
405 

1632  If 
164025 

F5939264 

C643J125 

20-099751-2 
20-1246118 

7-392542 
7-398636 

471  221841  1044871111  21-7025344 
472  222784!  105154048!  21-7255510 

7-780490 
7-785993 

406 

164836 

66923416)  20-1494417 

7  "104721 

473|  2237291  105823817  21  -74850*2 

7-79148? 

0? 

165  49 

67419143 

20-1742410 

7-410795 

474J  224676  106496424 

21-7715411 

7-790974 

08 

1564641  67917312 

20-1990099 

7-416859 

475!  225625 

107171875 

21-7944947  7-80245- 

409 

167281   68417929 

20-2237484 

7-422914 

476  226576 

107850176 

21-8174242)  7-807925 

410 

168100!  68921000 

20-2484567 

7-428959 

477 

227529 

108531333 

21  84032  J7  7-313389 

411 

168921  69426i  31 

20-2731349 

7-434994 

47fc 

223484 

109215352 

21-8632111  7-818846 

412 

169744 

69934J  23 

20-2977831 

7-441019 

47S 

229441 

109902239 

21-8850686  7-82429- 

413 
411 

170569 

171396 

70444997 
70957944 

20-3224014 
20-3469899 

7-447034 
7-453040 

430 
431 

230400 
231361 

1  105920UO 
111284641 

21-908J023  7-829735 
21-9317122!  7-H:«16j 

415 

172225 

71178375 

20-3715488 

7-459035 

432 

232324 

111930168 

21-9544934 

7-840595 

41- 

173055 

71991296 

20-3960781 

7-465022 

433 

233289 

112678537 

21-9772610 

7846013 

41? 

173389 

72511713 

20-4205779 

7-470999 

484 

234256 

113379J04 

22-OOOuOOO 

7851424 

418  174724 

73034632 

20-4450483 

7-476966 

4S5  235225 

114084125  y-2'0227155 

7-356823 

419!  175561 

73560059 

20-4694895 

7-482924 

486  236196  114791256 

220454077 

7-862224 

420i  176400 

74083000 

20-4939015 

7-488872 

487 

237169 

115501303)  ^-0680765 

7-867613 

421 

177241 

74618461 

20-5182845 

7-494311 

488 

238144 

116214272  22-OJ07220 

7-H72994 

422 

178084 

75151448 

20-5426336 

7-500741 

439 

239121 

116930169  22-1133444 

7-878368 

423  178929 

75636967 

20-5669638 

7-506661 

490 

240100 

117649000  22  1359435 

7-883735 

424!  179776 

76225024 

20-5912603 

7-512571 

491 

241031 

118370771!  22-1585193 

7-839095 

425  180625 

76765625 

20-6155281 

7-518473 

492 

242064 

119095438'  22-1811)730 

7-894447 

426  181476 

77303776 

20-6397674 

7-524365 

493 

243049 

119823157:  22^035033 

7-899792 

427t  182329 

77854483 

20-6639783  7-530248 

494 

244036 

120553784  222251KH 

7-905129 

428 

183184 

78402752 

20-6881609  7-535122 

495 

245025!  121287375J  22^35955 

7-910460 

429 

1840  11 

78953539 

20-7123152  7-541987 

4% 

246010!  122023936  222710575 

7-915733 

43li 

184900 

79507000 

20-7364414 

7-547842 

497 

247009 

122703473  222934963 

7-921099 

431 

185761 

80062991 

20-7605395 

7-553639 

493 

243004 

1235059  M  22-3159135 

7-926403 

432  1850*24 

80621568! 

20-7846097 

7-559526 

499 

249001  124251499)  22^333^79 

7-931710 

433,  187489 

81182737 

20-8086520 

7565355  500 

250000  125000000  22  360o798 

7-937005 

434  183356 

81746504 

20-8326667  7571174 

501 

251001 

125751501  22333J293 

7-942293 

435;  189225 

82312875 

20-8566536  7'576985 

502 

252004 

126506008  224053505; 

7-947574 

436  190096  82881856 

20-8806130|  7-582786 

503 

253009  127263527)  22-4270615) 

7-952848 

437  190969 

83453453 

20-9045450 

7-583579 

504 

254016  1280^4004  224499443 

7953114 

433  191844 

84027672 

20-9284495 

7-594363  505  255025  12878702  >i  22-472205  ll 

7-963374 

439  192721  84604519 

20-9523268!  7600133;  506 

250036;  129554216  22-4944433 

7-968627 

440  193600  85184000 

20-9761770 

7-005905  507 

257049  1303233431  22  516651)5 

7-973373 

441!  194481  8576)121 
442;  195364  86350388 

21-0000000  7-611663  508 
21-0237960)  7-617412  509 

253064  131090512!  22  5333553' 
259031  131872229)  225510243 

7-9791  U 
7-934344 

443!  196249 

86938307 

21-0475652!  7-623152  510!  2601001  132651000)  22'5331795| 

7-989570 

444  197136  87528334 

21-0713075!  7-023384  511 

261121  133432:131!  22-6053091! 

7-994788 

445  198025  88121  125  ! 

21-0950231 

7-63460?  512 

202144  134217728  22-0274  17o! 

8000000 

446!  1939161  88716536 

21-1187121 

7010321  513 

263169  13500569?)  22-6195033 

8-00521  * 

447 

199809;  89314623 

21-1423745  7010027)!  514 

264190  1357:Mi74l 

220;  15531) 

448 

2007041  89915392 

21-1660105  7-051725!  515 

265225  136590875 

22-6936  1  1  4 

8-0155'JJ 

449  2016011  90518349 

21-1896201 

7657414  510 

266256  137333096 

227156334 

8-020779 

450  202500!  91125000 

212132034  7-663094  517 

2672891  13318*413 

22-7376340!  »  02595? 

i:-l  203401  91733851 

21-2367606  7-663760  5  Is 

263324  138991832 

22-75J6134) 

8-031129 

452  204304  92345403 

21-2502916  7-071430-  519 

269351S  139?'.K!.-.' 

22-7815715 

8-03529/ 

453  305209)  92959677 

21-2837967;  7'080036  520 

•270400  1406U8000 

22-8u350a5 

M-041451 

454  206116  93576664 

21-3072758'  7'685733  521 

271441  141420761  228254244 

8-040003 

4531  207025  94196375 

21-3307290  7'69137'2  52'2 

•272  4^1  142236648 

22-847311*3 

8-051748 

456  21)7936  94818816 

21-3541565;  7  6J7002  523 

•273529  1  UUV.OO; 

22-3691933 

8  056336 

457!  208349  95443993 

21-3775533  7'702025  524 

274576  143377824 

22-89104  631 

8062018 

458;  209764 

96071912 

21-4009340  7-708239  5J5 

2750-25  1417.131,5 

•2-2-9128785 

8-067143 

459i  210581 

96702579! 

21-4242353  7713S15  526 

276676  145531570 

22-93  10  ^99 

8-07-2262 

460  211600 

97336000 

2  1-4  176  UK;  7-719413  5-27 

277729  145363183 

22-95.i4S.j6 

8077374 

461  212521 

97972181 

21-4709  UK")  ?-725:>32  528 

278784  147197952 

22-973^500 

8-082430 

462  213444 

ys.il  1123 

21-4941853  7  730014  529 

279341  148035339 

•2300JOJOO, 

463  214369 

99252347 

21-51743J8  77361HH  -,30 

280900  148877000 

23-0217.^9 

809267* 

464  215296 

99897344 

21-5406592  7711753  531 

281961J  149721291 

•23(1134372 

8-097759 

465!  216225  1005!  1625 

21-56335871  7'7473lir  532 

283024!  150558763 

23-065  1  -2  J  2 

8-102339 

466;  217156)  101194696 

21-5370331 

7-752861  533 

284089  15  14  1'9  i:i7 

23080792.3 

8-107913 

46?  21&J39J  101847563 

21-6101828 

7-758402  531 

285156  152273304 

23  1084400 

B'ltsfctiO 

46S  2190241  102503232 

21-0333077 

7-703.-30  5!55 

286225  153130375 

23  13,  "0070 

8-118041 

469  219961  10)161709 

21-6564078 

7-769462  536 

287296  153990656  23  1516738 

642 


APPENDIX. 


No. 

Square. 

Cub*. 

Sq.  Root. 

CuheRoot-'  No.  Squar,;. 

Cube.   1  S.].  Rcu. 

CutijKoot. 

537 

2333691  154854153  23-1732605'  8-123145J  604 

3-4816 

220348864 

245764115 

8  453026 

£33 

239444|  155720872;  23-1948^70  8-133187|  605 

3;.  6025 

221445125 

24-5967478  8-457691 

539 

2935211  156590819;  23-2163735  8-1332231 

606 

3<>7236 

222545016 

24-6170673!  8-462318 

540  2916001  157464030;  23-23790011  8-143253 

607 

368449 

223648543 

24-63737001  8-467001) 

541  292631*  158340421;  23-2594067.  8-148276 

608 

369664 

224755712 

24-6576560 

8-471647 

542  293764',  1592201)881  23-2H03935!  8-153294 

609 

370881!  225866529 

24-6779254 

8-476289 

543|  294349'  160103007  23-3023604!  8-158305 

610 

372100]  226981000 

246981781 

8-480926 

544  295936 

160939184  23-3233076  8-163310!  611 

373321 

228099131 

24-7184142 

8-485558 

545  297025 

161873625  23-3452351 

8-168309  612 

374554 

229220928 

24-7386338 

8490185 

546  298116 

162771336J  23-3665429 

8-173302  613 

375769 

230346397 

24-7538358 

8-494806 

547J  299209 

163667323;  23-3330311 

8-178239 

614 

376996 

231475544 

24-7790234 

8-499423 

548,  300304 

1645665921  23-4093998 

8-183269 

615 

373225 

232608375 

24-7991935 

8-504035 

5  19|  301401 
.550  302500 

165469149 
166375000 

23-4307490 
23-4520788 

8-188244 
8-193213 

616 
617 

379456 
330689 

2337448961  24-8193473 
234885113  24-8394347 

8-508642 
8-513243 

551 

3J3601 

1672841511  23-4733392 

8-198175)  618 

331924 

236029032!  24-8596058 

8-517840 

552 

304704!  163196608 

23-4946802 

8-2031321  619 

333161 

237176659  24-8797106 

8-522432 

553  3J5809 

16911237.7 

23-5159520 

8-203032  620 

334400 

238328UOO]  24-8997992!  8-527019 

554  306916 

170031464 

235372046 

8-213027  621 

385641 

239433061  24-91987161  8-531601 

555  303025 

170953375 

235534333 

8-217966  622 

386884 

240641848 

24-9399278  8-535178 

556  309136 

171879616 

23-5/96522 

8-222893J  623 

38^129 

241804367 

24-9591679  8-540750 

557 

310249 

172308693 

23-6003474 

8-227825 

i  624 

339376 

242970624 

24-9799920  8'5453l7 

558 

311364 

173741112 

23  6220236 

8-232746 

625  !  390625 

244140625 

25-0000000  8-549880 

559  312431 

174676879!  236431808]  8-237661 

626  1  391876 

245314376 

25-01-99920  8-554437 

560  313600 

175516000  23-6543191 

8-242571 

627  393129 

246491833 

25-0399681 

8-553990 

561  314721 

176553481 

23-6854336 

8-247474 

623  394334 

247673152 

25-0599282 

8-563538 

562!  315344 

177504328 

23-7065392 

8-252371 

629  395641 

248853189 

25-0793724 

8-568081 

563  316969 

178453547  23-7276210 

8-257263 

630 

396900 

250047000 

25-0993003 

8-572619 

564  318096 

17941)6144  23-7481x312  8^62149 

631 

393161 

251239591 

25-1197134 

8577152 

565!  319225 

180362125  23769728.1 

8-267029!  632 

3J9424 

252435968 

25-1398102 

8-581681 

566  320356 

1813214%  23-7J07515  8-2719041!  633 

40068^ 

253638137 

25-1594913 

8-536205 

567  321489 

182284263 

23-81176  18|  8-2767731'  634 

401956 

254810104  25-1793566  8-530724 

568 

322624 

183250432 

23-83275:;6:  8-281635!  635 

403225 

256047375  25-1992C63J  8'595233 

569 

323761  184220009 

33*537409  8-236493  636 

404  »96 

257259456 

25-2190404  8'599748 

570 

324900  185193000 

23-8746728!  8-2  9  1344 

637 

405769 

258474853 

25-2333539  8'f>042V2 

571 

326011!  186169411 

238.156063  8-2J6190 

638 

407044 

239694072 

25-25366191  8-608753 

572 

327184 

187149248 

23-9165215  8-3J1030 

639!  408321 

260917119 

25  2731493  8-613248 

573 

328329 

183132517 

23-9374184  8-3J5865  640  409630 

262144000 

25.2932213!  8-617739 

574 

3294761  189119224 

23-9532  J71  8-310694 

641  410381 

263374721 

25-3179778J  8'622225 

575 

330625  190109375 

239791576  8-315517 

642J  412164 

264609283 

•25-3377189  8-626706 

576  331776  191102976 

24-0000030  8-3203M31  643 

413419  265847707 

25-35741471  8-631183 

577!  332929 

192103033 

24-0208213 

8-325147!  64-1 

414736 

267089984 

25-3771551 

8-635655 

578  334034 
579!  335241 

193100552 
194104539 

21-0416306 
24-0524183 

8-329954  645  416025 
8-33475J;;  616:  417316 

268336125 
269586136 

25-3968502  8.640123 
25-41653.il  8-6445S5 

580|  336400  195112000 

24-0331891  8-339551  !  647  418509 

270340023  25  -4  3,5  1947 

8-649044 

5811  337581,  196122941 

24-1039416  8-344341  648;  419904 

272097792  25  '45581411  8-653497 

582)  333724!  197137368 

24-1246762!  8-349126!  649  421201 

2733594491  254751784:  8-657946 

5331  339839 

193155237 

24-1453929  8-353905i  650i  422500 

274625000  25'4950976i  8-662391 

534J  341056  199176704 

24-1660919 

8-353678  65li  423301 

275894451  25-5147016:  8-666331 

535  342225!  200201625 

24-18(57732 

83534471  652!  425104 

2771(57808 

25-53429071  8-671266 

586  343395  201230055 

242)74359 

8-333209!  653  425409!  278445077 

25-5533647!  8  675697 

587  3445691  202262003 

24-2230829 

8-372967:  654  427716!  2*79726264!  25'5734237l  8-680124 

538  3457441  203297472 

242487113  8-377719:  655  4290251  281011375!  25'5929678;  8-68454f 

539J  346921  204335469 

24-2593222  8-332465 

1  656  43,336  282300416  25*6124969  8-683963 

590  343100,  20537'JOOOJ  242399156  8-337206  657i  431619!  2835933931  25-6320112  8693376 
591  349281  206425071  24-3104916  8-3919421'  658'  432964  234890312  25'6515107  8-697781 

*92'  350464:  207474683  213310501  8-3J6673!  659<  4342811  236191179!  25'6709953  8-702188 

593!  3516491  208527357 

24-3515913  8-4013981  660i  435600!  287496000!  25-6904652;  8-706538 

594i  352836  209584584 

24-3721152;  840611$,!  661|  436921 

288804781  25*7099203  8-710988 

595 

354025  210644875 

24-3926218!  8-410333J  6621  438244 

290117528  25-7293607!  8-715373 

596 

355216!  211703731 

24-4131112  8-415542!  663  439569 

291434247  25'7487854  8-719760 

59? 

356409  212776172 

24-4335834  8-420246!  664  440896 

292754944:  25'7681975  8724141 

59$ 

357604  213347192 

24-45403851  8-424945  665  442225 

294079625!  25'7875939!  8-728518 

59£ 

353801!  21492179ii 

21.4744765  8-429633!  666  443556 

295408296;  25-8069758'  8-732392 

601 

360001 

216000001 

244948974 

8-4343271;  667  444889 

296740963!  258263431;  8-737260 

GO] 

361201 

217081801 

24-5153012 

8-439010!  i  668  446224  298077632;  25  84561J6J!  8741625 

60S 

362404  21816720? 

2  4  -5356  -HI 

8-443688!'  669,  447561  299418309  25-8(5503431  8-745985 

605 

363609;  21925622? 

24-5560532 

8-448360J  670  448900  300763000!  25  "884  35  821  8-753340 

TABLE   OF   SQUARES,    CUBES,   AND    ROOTS. 


643 


No, 

Square. 

Cube.   |  Sq.  Root.  CubeRoot. 

No. 

Square. 

Cube. 

Sq.  Ro<t.  CubeRool. 

671 

450241 

302111711 

25-90366771  8-754691 

738 

544644  401947*72  27-16615541  9-036880 

672 

451534 

303464448 

25-9229623!  8-759033 

739 

546121 

403533419  27-1845544'  9  040965 

673 

674 

452929 
454276 

304821217 
306182024 

25-y422435|  8-763331 
25-9615100  8-767719 

740 
741 

547600 
549031 

405224COU  27-2029410!  9'04504* 
4U6369021  27-2213152  9-049114 

675 

455625 

307546375 

25-98076211  8-7720531  742 

550564 

408518488  27-23J6769  9'053l83 

676 

45b'97<i 

308915776 

26-0000000'  8-776333'  743!  552J49 

410172407  27-2530263'  9-057245 

677 

453329 

310288733 

26-0192237  878J708  744  553536 

411830784  27-2763634 

9061310 

678 

459684  311665752 

26-03813311  8-7850301  745!  555025 

413493625;  27-2946881 

679 
630 

461041 
462401. 

313046839 
314432000 

26.0576284J  8-789347 
26-07630%!  8-793659 

746  5565  lb 
747|  5J80(,9 

415160936!  27-31300L6 
416332723!  27-33130i)7 

9-073473 

681 

463761 

315821241 

26-0959767  8-797968 

74UJ  5J9504 

4185U8992 

27-34y5337 

y0775*0 

632 

46-H24 

bl7214568 

26-1151297  8-802272 

749  561001 

42Jl8974y 

273670644 

683 

466489 

313611987 

26-13426871  8-8U6572 

1  750  5625  JO 

42187.XXX) 

27-336127J 

y-085bOb 

684 

4678561  320013504 

26-1533937 

8-810868!)  7511  564o01 

4*3564751 

27-40437y2 

9-08^63j 

685 

469225 

321419125 

26-1725047 

8-815160 

752  565504 

4*5^59008 

27-4226134 

y-0j36<2 

686!  470596 

322828356 

26-1916017 

8-819447 

753 

o.j/uoy 

426957777 

27-4403455 

9-Oi)770l 

<>87i  471961) 
688J  473344 

3*4242703 
32566(;672 

262106848 
26-2297541 

8-823731 
8-823010 

754 
755 

568516 
570025 

4*^661064 
430J68375 

2745906o4 
27-47/2633 

9-l0172b 
9-105740 

609 

474721 

327082769 

26-2483095 

8-832285 

756 

571536 

432084215 

27-4^54542  9-109767 

690 

476100 

323509UGO 

26-*673511 

8-836556 

757 

573U49 

433798093 

9-113702 

691 

477431 

329939371 

26-2868789 

8-840823 

758 

574564 

43551951* 

2/-5317yy8 

y-l!77»3 

692 

478361 

331373888 

26-3053929  8-845085 

759 

57608  1 

437*45479 

275499546 

9-121801 

693 

480249 

33*2812557 

26-3248932!  b'Ws>344ll  760 

577600 

438976000 

2  T  -5680975 

y-  125805 

<94 

481636 

334255384 

26-34387^7 

8-853598  761 

57^121 

440711081 

27'53b2234 

9-12y8b6 

695 

483J25 

335702U75 

26-3628527 

8-857849  762 

530644 

442450728 

27-6043475 

y-13380b 

696 

484416 

337153536 

26-3318119 

8-862095  763 

532469 

444194947 

276224546 

y-  1377^7 

697 

485809 

338608373 

26-4007576 

8.866337  764 

533696 

4459437^4 

9-441787 

698 

48720-i 

3400GS392 

26-4196896 

8-8705761  765 

585225 

447697425 

^7  'Ojt^o33-i 

9-  44577  1 

699 

488601 

341532099 

26-4386081 

8-874810  766 

586756 

44*4550b6 

27  b7o705b 

9-l4975o 

700 

49UL'00 

343000000 

26-4575131 

8-879040  767 

533^09 

45121.663 

*7'694764o 

y-45o737 

701 

491401 

344472101 

*6-4764046 

8-883266 

768 

589824 

452984332 

*7-7l23i*y 

y-.'5/Vi^ 

702 

492i04 

345948408 

26-4952826 

8-887483 

769 

591361 

454756609 

27-730849* 

y-lblbo/ 

703  491209 

347428927 

26-5141472 

8-891706 

770 

592900 

456533000 

27-7480739 

9-165656 

704 

495616 

348913664 

26  5329983 

8-895920 

771 

594441 

45334401  1 

27-7663860 

9'169b2.i 

705 

497025 

b5u40~625 

26-5518361 

8-900130 

772 

5J5984 

4b009b648 

9-17353J 

706 

498436 

351895316 

26-57C6605 

8-904337 

773 

597529 

46183i)9i7 

278023775 

9-17/014 

707 

499849 

353393243 

26-5394716 

8-908539 

774 

599076 

463684824 

27  "8200555 

y  401500 

708 

501264 

354894912 

26-6082694 

8-912737 

775 

600625 

465481375 

27  -0  33021  8 

y  105453 

709 

502681 

356400829 

266270539 

8-916931 

776 

602176 

4b7*8e5/6 

27  o5b776o 

9  18940^ 

710 

504100 

357911000 

26-6453*52 

8-921121 

.777 

603729 

461J097433 

27-8747197 

y  I9b347 

711 

5U5521 

35942543! 

266645333 

8-925308 

778 

605284 

47U9l09;)2 

27'0b26ol4 

'j-197~9c. 

712 

5t6944!  360944128 

266833231!  8-929490 

779 

606841 

47V729139 

27-9ib57J5 

V*0l22lj 

713 

500369  362467097 

26-7020598 

8-9b3669 

780 

6034  b 

4/4552UOO 

27-9284801 

>j-2Ujlt>« 

714 

5097^6:  353994344 

26-7207784 

8-937843 

781 

60^961 

47t  37^541 

27-y4b3772 

y^o^o-Jb 

715 

511225!  365525375 

5T  -739483  J 

8-942014 

782 

6115^4 

47o^ll768 

27-9642629 

y-*i3o~o 

716 

512656  367061696 

26-7581763 

8-946181 

783 

613089 

40X04060V 

27'i»821b7*!  y-*ibi)Jv, 

717 

514089  368601813 

26-7763557!  8-950344 

;  784 

614656 

481890^04 

23-OCOOObO 

y-*2O8/o 

718 

515524  370146232 

26-7955220  8-9545o3|:  785 

616225 

40o/b66*5 

28  Ol7o5l5 

>j-»2l7v-l 

719 

516%l|  37169495U 

26-81447.-)! 

8-958653  706 

617796 

4855o'/b5o 

28  "0356  yl  5 

y-*2o7^/ 

720 

518400;  373248UOO 

26-8328157 

8-962809!  787 

61936'J 

407443-lob 

28-0535*00 

•.'•_.  'Jo  i'.. 

721 

519341  37181)5361 

26-8514432 

8-966957 

i  788 

620944 

48X)3o3i72 

28-071  33  // 

y-2ot5^6 

722  521234  37t.3,'.70v- 

26-8700577  8-971101 

789 

62*521  4yil6906'j 

28-089  143^ 

y-24o4b;> 

723  522729  377933i>r>7 

26-8886593  8-975241 

i  790 

6*4100  4y3b3yOoO 

23'106y38b 

9-*14b3o 

724!  524176  379503424 

26-9072481 

8-979377 

i  7<J1 

625631 

494940671 

if8'12472*2 

y-*48*b4 

725  525625  381078125 

26-9258240 

8-983509 

i  792 

627264 

49b'/'yi)oM^ 

23'l42494b 

y-2o2i3t 

726  527076  382657176 

*6-9  143872  8'9876:>7  793 

628849  4906/7*57 

23-1602557 

y-^obo2* 

727  528529  38424053« 

2-3-9629375  8-99176^  7J< 

63Ji:-.t}  o005b6404 

28'170bOJb 

9-*j99ll 

728!  529984'  38582835:. 

.6-98  14751  8-995883|:  ?y5 

6320*5  50*459375 

28-  495  i  444 

y-*6.uy< 

729  531441  3^7420489 

27-000(K)00  y-COOOOb 

?y6 

6336  1<) 

5D4353336 

20-2134720 

y-2b7bOv 

730  532900  38^0171-00 

27-0185122  9-004113 

79. 

635*0'J 

50626l57«> 

28-2311004 

'.I  -.  l-'-'v 

731  534361;  39C6l7s.ll 

270370117:  9-008223 

I  798 

636oOi 

5084bu5»* 

28'248893o 

y-^ij4oo 

7321  535324  392223168 

:.7-c<:>.V;985  9-012329 

1  7yy 

638401 

5iot8*3yy 

28*005801 

9-i7y3oc 

733!  5272*9  39383^837 

27-G73V727  9-016431 

>  800 

64000C 

512000000 

28-204271* 

y-2o3i7o 

734!  53/7;»6  3.i54469i»! 

27-0924341  'J  -02052'. 

801 

641604J  513922401 

28-3019434 

y-2a  7o-*-» 

735  :  540^25  3J7065375 

27.1108834  9-021621 

802 

643204  51534l6bO 

20-3  11.*  oi.) 

9-290.0* 

736  541*i»>  39'0,s(su;>(i  271293199  9-0*a7l5 

i  803 

644809  51778162/ 

28-o3/*54b 

9-*'j4<b< 

737,  f>431t.9  40031555:>  27  147743.*  9-032802 

804]  64641b 

5197181b4 

*tt-354oy30 

y-2y8ti*i 

644 


APPENDIX. 


No. 

Square. 

Cube. 

Sq.  Root. 

CubeRootJ 

No. 

Square.)   Cube. 

1 

Sq.  Root  CubeKoot. 

805 

6430251  52166J125 

2837*5219  9-302477 

872|  761)334 

663.)M348|  29-5296461 

9-553712 

806 

649636  52360G616: 

283901391 

9-306323 

873  762129 

665338617!  29-5465734 

9-557363 

807 

651249  525557943; 

28-4077454 

9-310175 

874 

763376 

667627624  29-5634910 

9-561011 

808 

652864 

527514112 

28  4253408 

9-3140191 

875  765625 

669921875 

29-5303989 

9-564656 

809 

654481 

529475129 

28-4429253 

9-317860; 

876 

767376 

<>?2221376 

29-59729721  9-568298 

810 

656100 

531441000 

28-4604989 

9-321697  877 

769129 

674526133 

29-6141853  9-571938 

811 

657721 

533411731 

28-4780617 

9-325532 

878  770884 

676836152 

29-63106481  9-575574 

812 

659344 

535387328 

23-4956137 

9-32J363 

879  772641 

679151439 

29-6479342,  9-579208 

813 

660969 

537367797 

23-5131549 

9-333192 

880  774400 

681472000 

29-6647939!  9-582840 

814 

662596 

539353144 

23-5306852 

9-347017 

b81  776  Hi  1 

683797341 

29-6816442  9-:86468 

815  664225 

541343375 

23-5482048 

93408391 

832  777924 

686128968 

29.6984843 

9-590094 

816 

665356 

543338496 

23-5657137 

9-344657! 

883  779689 

688465387 

29-7153159  9-593717 

81? 

6l>7489 

545333513 

28-5832119 

9-3:8473 

834  781456 

690807104 

29-7321375'  9-597337 

818 

669124 

547343432 

23-6006993 

9-352286 

8b5j  783225 

693154125 

29-7439496!  9-600955 

819 

670761 

549353259 

28-6181760 

9-356095 

886  784996 

695506456 

29-7657521  9-604570 

820 

672400 

551368000 

28-6356421 

9-359902! 

88? 

786769 

697864103 

29-7825452 

9-603182 

821 

674041 

553387661 

28-6530976 

9363705 

838 

788544 

700227072]  29-7993239 

9-61*791 

822 

675684 

555412248 

28-6705424 

9367505: 

889 

790321 

702595369 

29-816103:) 

9-6153'J'r 

823 

677329 

557441767 

23-63797G6 

9-371302 

890 

71J2100 

704969000  29-8323678 

9-619002 

824 

678976 

559476224 

23-7054002 

93750961 

891 

793881 

707347971  29-8496231 

9-622603 

825 

680625 

561515625 

28-7228132 

9-373387 

892 

795664 

709732288  29-8663690 

9-626202 

826 

632276 

563559976 

28-740215? 

9-382675 

893 

797449 

712121957  29-8831050 

9-629797 

827 

633429 

565609283 

28-7576077 

9-336460 

894 

799236 

714516934 

29-8998328 

9-633391 

823 

685584 

567663552 

28-7749891 

9-390242 

895 

801025 

716917375  29-9165506 

9-636981 

829 

687241 

569722789 

28-7923601 

9-394021! 

896 

802816 

719323136  2l  ^332591 

9-640563 

830 

6839  >0 

571787000 

28-8097206 

9-3.  7796! 

897 

804609 

721734273 

29-9499583 

9-644154 

831 

690561 

5?385619l|  28-8270706  9-401569 

898 

806404 

724150792 

29-9666481 

9-647737 

832 
833 

69i224 

693889 

5?5930368|  28-8444102;  9-405339 
5780095371  28-86173941  9-409105 

899 
900 

808201  726572699 
810000|  729000000 

29-98332-37  9-65131? 
30-0000000!  9-654894 

834 

695556 

530093704 

28-8790582!  9-412869 

901 

811801  731432701 

30-0166620  9-658468 

835  697225 

582182875 

28-8963666!  9-416630 

902 

813604!  733870808 

30-0333148  9-662040 

836:  698896 

584277056 

28-9136646  9-420337 

903 

815409 

73631432? 

30-0499584  9-665610 

837  700569 

586376253 

28-9309523 

9-424142 

904 

817216 

738763264 

30-0665923  9-669176 

838  i  702244 

5^8480472 

28-948229? 

9-4278J4 

905 

819025 

741217625 

30-0332179 

9-672740 

839  703921 

590589719;  23-965496? 

9-431642 

906 

820836 

74367741<i 

30-0998333 

9-676302 

840!  705600 

592704000  28-9827535  9-435338 

90? 

822649 

746142643 

30-1164407 

9-679860 

841  707281 

594823321 

29-OOOOOOC 

9-439131 

908 

824464|  748613312 

30-1330383 

9-f>83417 

842J  708^64 

596947688 

29-0172362 

9-442370 

909 

826281 

751089429 

30-149626'J 

9-036970 

843  710649 

599077107 

29-0344622 

9-446607 

910 

828100 

753571000 

30-1662063 

9-690521 

844j  712336!  6012  1  1584 

29-0516781 

9-450341 

911 

829921 

756058031 

30-1827765 

9-694069 

8451  7140251  603351125 

29-0688837 

9-454072 

912 

831744 

758550528 

30-1993377 

9-697615 

8461  715716J  605495736 

29-0860791 

9-457800 

913 

833S6S 

761048497 

30-2158393 

9-701158 

847  717409  607645123  29-1032644 

9-461525 

914 

835396]  763551944 

30-232432b 

9-704699 

84fc 

719104 

609300192)  29-1204396 

9-465247 

915 

8372251  766060875 

30-248966S 

9-708237 

84£ 

720801 

611960049  29-137604f 

9-463966 

916 

839056  768575-296 

30-26549  It 

9-711772 

85C 

>  72250C 

614125000  29-154759= 

9-47*682 

917 

840889  771095213 

30-282007L 

9-715305 

851!  724201 

6162950511  29-  17  19042 

t  9-476396 

918 

842724 

773620632 

3J-2(J8514fc 

9-718835 

852!  725904 

618470208!  29  1890390  9-480106 

919 

8445611  7761515511 

30-3150  12b 

9-722363 

853  72760'j 

6206504771  29-2U61637J  9-483814 

92u 

84640C 

1  773683000 

30-3315016 

9-725388 

854 

7293  If 

622835364  29-2232784  9'48?518 

921 

848241 

781229961 

30-347981^ 

9-729411 

85f 

7310^' 

625026375  29-2403830  9-491220 

922J  850084  78377744ti 

3J-364452L 

9732931 

856 

,  73273f 

627222016  29-2574777  9-494919 

923  851929  786330467 

303809151 

9-736448 

85' 

'  73444i 

629422793J  29-2745623!  9-498615 

924 

853776:  78388902-4 

30-3J7363? 

9-73J963 

85£ 

J  736164 

631623712!  29'2916370|  9-5U23J8 

925 

855625!  791453125 

30-4138127 

9-743476 

85i 

>  737881 

633339779  29-3087018!  9-505998 

926 

857476  79402277f 

3J-43J2481)  9-746986 

860  73960C 

6360560(X)|  29  3257566  9-509685  927 

859329  796597982 

3J-4466747  9-750493 

861  74132] 

6382773811  29-3423015J  9-513370 

928 

861184 

79917875. 

3U-4630924I  9-753J98 

862  743044 

640503928  29-3598365  9517051 

92'J 

fc63041 

801765031 

30-47y5013|  9-757500 

863  74476i 

6427356471  29-3768616  9-520730 

931) 

864900  804357000  30-4i/59014|  9-761000 

8641  746496!  644972544!  29-3933769  9-524406 

931 

866761'  806954491  30-5122926J  9-76449? 

865  748225  6472146251  29-4103823  9-528079 

932 

868624.  8095575681  30-5285750  9-767992 

866  749956  649461896  29-4278779  9-531750 

932 

87J4*-»j  812i6G237|  30-545048?  9-771484 

867  75168i 

>  651714363  29-4448637  9-535417 

934 

872356;  814780504  30-5:5  14  136!  9-?74i>74 

868;  75342^ 

[  653972032'  29-4618397  9-539082 

933 

874225!  817400375 

30-67776911  9-778462 

869!  75516] 

656234909  29-4788059  9-542744 

93fc 

876096:  820025y5f 

30-5;;4Ii71  978194? 

87C1  75690( 

)  658503000  29-4957624  9  "5  46403 

937 

877969:  82265615; 

3D-610455?;  9  785429 

87  1\  75864 

660776311!  29-51270'Jli  9-55UU5!) 

933  H79H44  825293675! 

30-6267857  9-736908 

TABLE  OF  SQUARES,  CUBES,  AND  ROOTS. 


645 


No. 
939 

Square. 

881721 

Cube.   |  Sq.  Root. 

CubeRoot. 

No. 

Square. 

Cube. 

Sq.  Root. 

CabeRooiJ 

827936019:  30-6431069 

9-792386 

970 

940900 

912673000 

3H448230J  9-8389831 

940 

883600 

830584000!  33-6594194 

9-795861 

971 

942341 

915498611 

31-160872'J 

9-902333 

941 

885481 

8332376211  30-6757233 

9-799334 

972 

944784 

918330048 

31-1769145 

9-905782 

942 

837364 

835896888  30-6920185 

9-802804 

973 

946729 

921167317|  31-1929479 

9-909173 

943 

889249 

83856  1807J  30-7083051 

9-806271 

974 

948676 

924010424!  31-2039731 

9-912571 

944 

8U1136 

8412323341  30-7245830 

9-809736 

975 

950625 

926859375J  31-2249900 

9-915962 

945 

893025 

843908625  30-7408523 

9-813199 

976 

952576 

929714176 

31-2409987 

9-919351 

946 

894916 

846590536  30-7571130 

9-816659 

977 

954529 

932574833 

31-2569992!  9-922733 

947 

896809 

849278123  30-7733651 

9-820117 

978 

956484 

935441352 

31-27299151  9-926122 

948 

898704 

851971392!  30-7896086 

9-823572 

979 

958141  938313739 

31-288.^757 

9-929504 

949 

900601 

8546703491  30-8058436 

9-827025 

980 

960400 

941192000 

31-3049517 

9-9328S4 

950 

90:2500 

8573750001  30-82207001  9-830476 

981 

962361  944076141 

31-3209195  9-936261 

951 

904401 

860085351  308382879  9'833924 

982 

964324!  946966168 

31-3368792  9-939636 

952 

906304 

862801408:  30-85449721  9-837369 

983 

966289  949862087 

31-3528308 

9943009 

953 

908209 

865523177  30-870698l'  9-840813 

984 

968256!  952763904 

31-36877431  9-946380 

954 

910116 

868250664  30-88689041  9'844254 

985 

970225  955671625;  31-3847097!  9-949748 

K55 

912025 

870983875  30-9030743,  9-847692 

986!  972196 

958535256 

31-400H369)  9-953114 

956 

913936 

873722816  30-9192497J  9-851128 

987;  974169 

961504803 

31-4165561  9-956477 

957 

915849 

876467493;  30-9354166!  9-854562 

2881  9761441  9644302"2 

31-4324673!  9-959839 

958 

917764 

879217912  30-95  15751J  9-857993 

989 

9781211  967361669 

31-4483704  9-963198 

959 

919681 

881974079  30-9f>77251|  9-861422 

990 

980100 

970299000 

31-4642654  9-96655.l> 

960 

921SOO 

884736000^  30-9^36668  9-864848 

991  982081 

973242271 

31-4801525  9-969909 

961 

923521 

887503681  31-OOOOOOOi  9-868272 

992  984064 

976191488 

31-4960315!  9-973262 

962 

925444 

NW277128  31-01612481  9-871694 

993i  986049 

979146657 

31-5119025 

9-976612 

963 

927369 

;<<.»:J056347  3l-0322413i  9875113 

994  988036 

982107784 

31-5277655 

9-979960 

964 

929296 

S;).r)841344  31-0483494J  9-878530 

995  990025 

985074875 

31-5436206 

9-983305 

965 

931225 

898632125  31-0644491  9-881945 

9% 

992016 

988047936 

31-5594677 

9-986649 

966 

933156 

90M28696  21-08054051  9-885357 

9971  994009 

991026973J  31-5753068 

9-989990 

967 

935089 

104231063  31-0966236  9-888767 

998  996004 

994011992 

31-5911330 

9-9933291 

968 

937024 

907039232  31-1126984  9-892175 

999 

998001 

997002999 

31-6069613 

9-996666, 

969  938961 

909853209  31-1287648!  9-895580  1000 

1000000  1000000000 

31  -6227766  lO-OOOOOOJ 

The  following  rules  are  for  finding  the  squares,  cubes,  and  roots  of  num- 
bers exceeding  1000. 

To  find  the  square  of  any  number  divisible  without  a  remainder.  Rule. — Di- 
vide the  given  number  by  such  a  number  from  the  foregoing  table  as  will 
divide  it  without  a  remainder  ;  then  the  square  of  the  quotient,  multiplied  by 
the  square  of  the  number  found  in  the  table,  will  give  the  answer. 

Examfle.—Whzt  is  the  square  of  2000  ?  2000,  divided  by  1000.  a  number 
found  in  the  table,  gives  a  quotient  of  2,  the  square  of  which  is  4,  and  the 
square  of  1000  is  1,000,000,  therefore  : 

4  x  1,000,000  =  4,000,000:  the  Ans. 

Another  Example. — What  is  the  square  of  1230?  1230,  being  divided  by 
123,  the  quotient  will  be  10,  the  square  of  which  is  100,  and  the  square  of  123 

is  15,129,  therefore  : 

100  x  15,129=  1,512,900:  the  Ans. 

To  find  the  square  of  any  number  not  divisible  without  a  remainder.  Rule. — 
Add  together  the  squares  of  such  two  adjoining  numbers  from  the  table  as 
shall  together  equal  the  given  number,  and  multiply  the  sum  by  2  ;  then  this 
product,  less  i,  will  be  the  answer. 

Example. — What  is  the  square  of  1487  ?  The  adjoining  numbers,  743  and 
744,  added  together,  equal  the  given  number,  1487,  and  the  square  of  743  = 
552,049,  the  square  of  744  =  553,536,  and  these  added  =  1,105,585,  therefore  : 

1,105,585  x  2  —  2,211,170  —  i  =  2,211,169 :  the  Ans. 

To  find  the  ciibc  of  any  number  divisible  without  a  remainder.  Ritle. — Divide 
the  given  number  by  such  a  number  from  the  foregoing  table  as  will  divide 


646  APPENDIX. 

it  without  a  remainder  ;  then  the  cube  of  the  quotient,  multiplied  by  the  cube 
of  the  number  found  in  the  table,  will  give  the  answer. 

Example. — What  is  the  cube  of  2700?  2700,  being  divided  by  900,  the  quo- 
tient is  3,  the  cube  of  which  is  27  and  the  cube  of  900  is  729,000,000,  there- 
fore : 

27  x  729,000,000  =  19,683,000,000  :  the  Ans. 

To  find  the  square  or  cztbe  root  of  numbers  higher  than  is  found  in  the  table. 
Rule. — Select,  in  the  column  of  squares  or  cubes,  as  the  case  may  require,  that 
number  which  is  nearest  the  given  number  ;  then  the  answer,  when  decimals 
are  not  of  importance,  will  be  found  directly  opposite,  in  the  column  of  num- 
bers. 

Example. — What  is  the  square  root  of  87,620?  In  the  column  of  squares, 
87,616  is  nearest  to  the  given  number ;  therefore,  296,  immediately  opposite 
in  the  column  of  numbers,  is  the  answer,  nearly. 

Another  example. — What  is  the  cube  root  of  110,591?  In  the  column  of 
cubes,  110,592  is  found  to  be  nearest  to  the  given  number  ;  therefore,  48,  the 
number  opposite,  is  the  answer,  nearly. 

To  find  the  cube  root  more  accurately.  Rule. — Select  from  the  column  of 
cubes  that  number  which  is  nearest  the  given  number,  and  add  twice  the 
number  so  selected  to  the  given  number  ;  also,  add  twice  the  given  number 
to  the  number  selected  from  the  table.  Then,  as  the  former  product  is  to  the 
latter,  so  is  the  root  of  the  number  selected  to  the  root  of  the  number  given. 

Example. — What  is  the  cube  root  of  9200?  The  nearest  number  in  the  col- 
umn of  cubes  is  9261,  the  root  of  which  is  21,  therefore  : 

9261  9200 

2  2 


18522     18400 
9200      9261 


As  27,722  is  to  '27,661,  so  is  21  to  20-953  +  ,  the  Ans. 

Thus,  27,661  x  21  =  580,881,  and  this  divided  by  27,722  =  20-953  -f . 

To  find  the  square  or  cube  root  of  a  whole  number  with  decimals.  Rule. — Sub- 
tract the  root  of  the  whole  number  from  the  root  of  the  next  higher  number, 
and  multiply  the  remainder  by  the  given  decimal  ;  then  the  product,  added  to 
the  root  of  the  given  whole  number,  will  give  the  answer  correctly  to  three 
places  of  decimals  in  the  square  root,  and  to  seven  in  the  cube  root. 

Example. — What  is  the  square  root  of  11-14?  The  square  root  of  n  is 
3-3166,  and  the  square  root  of  the  next  higher  number,  12,  is  3-4641  ;  the  for- 
mer from  the  latter,  the  remainder  is  0-1475,  and  this  by  0-14  equals  0-02065. 
This  added  to  3-3166,  the  sum,  3-33725,  is  the  square  root  of  11-14. 

To  find  the  roots  of  decimals  by  the  use  of  the  table.  Rule. — Seek  for  the 
given  decimal  in  the  column  of  numbers,  and  opposite  in  the  columns  of  roots 
will  be  found  the  answer,  correct  as  to  the  figures,  but  requiring  the  decimal 
point  to  be  shifted.  The  transposition  of  the  decimal  point  is  to  be  performed 
thus  :  For  every  place  the  decimal  point  is  removed  in  the  root,  remove  it  in 
the  number  t^vo  places  for  the  square  root  and  three  places  for  the  cube  root. 


THE   REDUCTION   OF   DECIMALS.  647 

Examples. — By  the  table,  the  square  root  of  86-0  is  9-2736,  consequently  by 
the  rule  the  square  root  of  0-86  is  0-92736.  The  square  root  of  9-  is  3-,  hence 
the  square  root  of  0-09  is  0-3.  For  the  square  root  of  0-0657  we  have 
0*25632,  found  opposite  No.  657.  So,  also,  the  square  root  of  0-000927  is 
0-030446,  found  opposite  No.  927.  And  the  square  root  of  8-73  (whole  num- 
ber with  decimals)  is  2-9546,  found  opposite  No  873.  The  cube  root  of  0-8 
is  0-928,  found  at  No.  800  ;  the  cube  root  of  0-08  is  0-4308,  found  opposite 
No.  80,  and  the  cube  root  of  0-008  is  0-2,  as  2-0  is  the  cube  root  of  8-0.  So 
also  the  cube  root  of  0-047  's  0-36088,  found  opposite  No.  47. 


RULES   FOR  THE   REDUCTION    OF   DECIMALS. 

«•, 

To  reduce  a  fraction  to  its  equivalent  decimal.  Rule. — Divide  the  numerator 
by  the  denominator,  annexing  cyphers  as  required. 

Example. — What  is  the  decimal  of  a  foot  equivalent  to  three  inches  ? 
3  inches  is  j\  of  a  foot,  therefore  : 

•&  .  .  .  12)3-00 

•25  Ans. 
Another  example. — What  is  the  equivalent  decimal  of  $  of  an  inch? 

1  ...   8)7-000 

•875  Ans. 

To  reduce  a  compound  fraction  to  its  equivalent  decimal.  Rule. — In  accordance 
with  the  preceding  rule,  reduce  each  fraction,  commencing  at  the  lowest,  to 
the  decimal  of  the  next  higher  denomination,  to  which  add  the  numerator  of 
the  next  higher  fraction,  and  reduce  the  sum  to  the  decimal  of  the  next  higher 
denomination,  and  so  proceed  to  the  last ;  and  the  final  product  will  be  the 
answer. 

Example. — What  is  the  decimal  of  a  foot  equivalent  to  five  inches,  f  and  TV 
of  an  inch  ? 

The  fractions  in  this  case  are,  i  of  an  eighth,  |  of  an  inch,  and  Trv  of  a  foot, 
therefore  : 


eighths. 

inches. 

•rV 12)5-437500 

-453125  Ans. 

The  process  may  be  condensed,  thus  :  write  the  numerators  of  the  given 


648  APPENDIX. 

fractions,  from  the  least  to  the  greatest,  under  each  other,  and  place  each  de« 
nominator  to  the  left  of  its  numerator,  thus  : 


8 

3.  CQOO 

12 

5-437500 

•453125  Ans. 

To  reduce  a  decimal  to  its  equivalent  in  terms  of  lower  denominations.  Rule. 
— Multiply  the  given  decimal  by  the  number  of  parts  in  the  next  less  denomi- 
nation, and  point  off  from  the  product  as  many  figures  to  the  right  hand  as 
there  are  in  the  given  decimal  ;  then  multiply  the  figures  pointed  off  by  the 
number  of  parts  in  the  next  lower  denomination,  and  point  off  as  before,  and 
so  proceed  to  the  end  ;  then  the  several  figures  pointed  off  to  the  left  will  be 
the  answer. 

•Example, — What  is  the  expression  in  inches  of  0-390625  feet? 

Feet  0-390625 

12  inches  in  a  foot. 


Inches  4-687500 

8      eighths  in  an  inch. 

Eighths  5  •  5000 

2  sixteenths  in  an  eighth. 

Sixteenth  i-o 

Ans.,  4  inches,  £  and  •£$. 

Another  example. — What   is  the  expression,    in  fractions  of  an  inch    of 
0-6875  inches? 

Inches  0-6875 

8  eignths  in  an  inch. 

Eighths  5-5000 

2        sixteenths  in  an  eighth. 

Sixteenth  i-o 

Ans.,  f  and  ^ 


TABLE   OF   CIRCLES. 

(From  Gregory's  Mathematics.) 

FROM  this  table  may  be  found  by  inspection  the  area  or  circumference  of  a 
circle  of  any  diameter,  and  the  side  of  a  square  equal  to  the  area  of  any  given 
circle  from  i  to  100  inches,  feet,  yards,  miles,  etc.  If  the  given  diameter  is  in 
inches,  the  area,  circumference,  etc.,  set  opposite,  will  be  inches  ;  if  in  feet, 
then  feet,  etc. 


Diam. 

Area. 

Circum. 

Side  of 
equal  sq. 

Diam. 

Area. 

Circum. 

Side  of 
equal  sq. 

•25 

•04908 

•78539 

•22155 

•75 

90-7625? 

33-7721-2 

9-52693 

•5 

•19635 

1-57079 

•44311 

11- 

95-03317 

34-55751 

9-74S1'.* 

•75|            '44178 

235619 

•66467 

•25 

99-40195 

35-34291 

9-97005 

]• 

•78539 

3-14159 

•88622 

•5 

103-86890 

36-12331 

10-19160 

•25 

1-22718 

3-92699 

1-10778 

•75 

108-43403 

36-91371 

10-41316 

•5 

1-76714 

4-71238 

1-32934 

12- 

11309733 

37-69911 

10-63472 

•75 

2-40528 

5-49778 

1-55089 

•25 

117-85831 

3848451 

10-85627 

2- 

3-14159 

6-28318 

1-77245 

•5 

122-71846 

3926990 

11-07733 

•25 

3-9760? 

7-06858 

1-99401 

•75 

127-676281       40-05530 

11-29939 

•5 

4-90873 

7-85393 

2-21556 

13- 

132-73228 

40-84070 

11-5-2095 

•75 

5-93957 

8-63937 

2-43712 

•25 

137-88646 

41-62610 

11-7425;) 

3- 

7-068581         942477 

2-65868 

-.-) 

14313881 

42-41150 

11-96406 

•25 

8-29576 

10-21017 

2-88023 

•75 

148-48934 

43-19689 

12-1856-2 

•5 

9-62112 

10-99557 

3-10179 

14- 

153-93804 

43-98229 

12-40717 

•75 

11-044661       11-78097 

3-32335 

•25 

159-48491 

44-76769 

12-62S73 

4- 

12-56637        12-56637       3-54490 

•5 

165-12996 

45-55309 

12-85029 

•25 

14-186251       13-35176J       3-76646! 

•75 

170-87318 

46-33849J     13-07184 

•5 

15-90431 

14-13716|       3-98802 

15- 

176-71458 

47-12338!     13-21)340 

•75 

17-72054 

14-922561       4-20957! 

•25 

182-65416 

47-909281     13-51496 

5- 

19-63195 

15-70796        4-431  13j 

-.r> 

188-69190 

48-69468      13-73651 

•25 

21-64753 

16-49336        4-65269J 

•75 

194-82783 

49-48003      13-95307 

•5 

23-75829 

17-27875 

4-87424 

IS- 

201-06192 

50-26548      14-17963 

•75 

25-96722 

18-06415 

5  -095  WO, 

•25 

207-:W4'20 

51-05088:     14-101  IS 

6- 

28-27433 

18-84955       5-31736 

•5 

213-82464 

51-83027      14-62274 

•25 

30-67961 

19-63495!       5-53891 

•75 

220-35327 

5262167 

14-84430 

•5 

33-18307 

20-42035       5-76047     17- 

226-98006 

53-40707 

15-065-»5 

•75 

35-78470 

21-20575.       5-9S2()3i'       -25 

233-70504 

54-19247 

15-28741 

7- 

33-48456 

21-99114 

6-20358!  !      -5 

240-52818 

54-97787      15-50897 

•25 

41-28249 

27-77654 

6-42514  j       -75 

•217-44950 

55763261     15-73052 

.K 

44-17H61 

23-56194 

6-646701:  18- 

254-46900 

56-54866!      15-9.V20S 

•75 

47-17297 

24-34734 

6-86825 

•25 

261-58667 

57-33406 

16-17361 

8- 

50-26518 

25-13274 

7-08981 

•5 

26880252 

53-11946 

1639519 

25 

53-45616 

25-91813 

7-31137 

75        27()-llC,54 

58-90486 

16-61675 

•5 

5S-74501 

26-70353 

7-53292 

19-           2-<3-52873 

59-69026 

16-8:KH 

•75 

60-13204 

27-48893       7754  18 

•25       291-03910 

6047565 

17-05986 

9- 

63617-2") 

28-27433       7-976J4 

•5  i       298-64765 

61-26105 

17-28  U2 

•25 

07-20063 

29-0.'>97*       8-197.')9 

•75 

300-:{513? 

62-04645 

17-5;»-2'J8 

•5 

70-83218 

29-84513       8-41915 

20- 

314-15926 

62-83185 

17-72453 

•75 

74-66191 

30-63052       8-64071 

"J.->        322-00-233 

6361725 

17-94609 

10- 

78-53981 

31-41592       886226 

•5 

330-06357 

64-40264 

18-16765 

•25 

82-51589 

38*01321      9-08382 

•75 

338-16-299 

65-18804 

18-38920 

•5           86-59014 

32-986721       9-3i):>33     21-     I       346-36059 

6  j  97341 

18-61076 

650 


APPENDIX. 


)iam. 

Area. 

Circam. 

Side  of 
equal  sq. 

Diam. 

Area. 

Side  of 
Circum.    enual  sq. 

21-25 

354-65635 

66-75834 

1883232 

38- 

1134-114941  119-38052!  3367662 

5 

363-05030 

67-54424 

19-05337 

•25 

1149-086601  120-  16591  i  33  893  17 

•75 

371-54241 

68-32964 

19-27543 

•5 

1164-15642 

120-95131   34-  IS  973 

22-     380-13271 

69-11503 

19-49699 

•75 

1179-32442   121-73671   3434129 

•25  1   388-82117 

69-90043 

19-71854 

39- 

1194-59060   122-52211 

34-56285 

•5 

337-60782!   70-68583 

19-94010 

•25 

1209-95495 

123-30751 

34-78440 

•75 

406-49263 

71-47123 

20-16166 

•5 

1225-41748   124-09290 

35-00596 

23- 

415-475^2 

72-25663 

20-38321 

•75 

1210-97818 

124-878301  35-22752 

•25 

424-55679 

73-04202 

20-60477 

40- 

1255-63704!  125-663701  35  44907 

•5 

433-736131   73-82742 

20-82633 

•25 

1272-39411 

126-449101  35-67063T 

•75 

443-01365|   74-61282 

21-04788 

•5 

1288-24933 

127-23450   35-89219 

24- 

452-38934J   75-39822 

21-26944 

•75 

:  1304-20273 

128-01990  36-11374 

•25 

461-86320 

76-18362 

21-49100 

41- 

1320-25431   123-80529J  36-33530 

•5 

471-43524 

76-96902 

21-71205 

•25 

1336-40406 

129-59069   36-55686 

•75 

481-105461   77-75441 

2193411 

•5 

1352-65198 

1  30-3760')  36-77841 

25- 

490-87385!   78-53981 

22-15567 

•75 

1368-99808 

131-16149   36-99997 

•25 

500-740411   79-32521 

22-37722 

42- 

1385-44236 

131-94689   37-2*2153 

•5 

.  510-70515!   80-11061 

22-59878 

•25 

1401-98480 

13273228   37-44308 

•75 

520-76306!   80-89601 

22-82034 

•5 

1418-625431  133-51768!  3766464 

26- 

530-92915   81-68140 

23-04190 

•75 

1435-36423 

134-303081  37-88620 

•25 

541-18842 

82-46680 

23-26345 

4o- 

1452-20120 

135-08348  38-10775 

•5 

551-54586 

83-25220 

23-48501: 

•25 

1469-13635 

135-87338   38-32931 

•75 

562-00147 

84-03760 

2370657 

•5 

1486-16967 

136-65928!  38-55087' 

27- 

572-55526 

84-82300 

23-92812; 

•75 

1503-30117 

137-44467J  33-77242 

•25 

583-20722 

85-60839 

24-14968 

44- 

1520-53084 

138-83007  38-993^8 

•5 

593-95736 

86-39379 

24-37124 

•25 

1537-85869 

1P9-01547J  39-21554 

•75 

604-80567 

87-17919 

24-59279; 

•5 

1556-23471   139-80087!  39-43709 

28- 

615-75216 

87-96459 

24-81435! 

•75 

1572-80890 

140-58627 

3965865 

•25 

626-79682 

88-74999 

25-03591 

45- 

152043128 

141-37166 

39-88021 

•5    637-93965 

89-53539 

25-25746 

•25 

1608-15182 

142-15706  40-10176 

•75 

649-18066 

90-32078 

25-47'J02 

•5 

1625-97054 

142-94246!  40-32332 

29- 

66051985 

9M0613 

25-70058 

•75 

164388744 

143-727861  4054488 

•25 

671-95721 

9  39153 

25-92213 

46- 

1661-90251   14451326!  40-76643 

•5 

683-49275 

92-67698 

26-14359 

•25   1680-01575   145'29866l  40-98799 

•75 

695-12646 

93-46233 

26-36525 

•5   1698-22717 

146-08405  41-20955 

30- 

706-85834 

94-24777 

26-58680 

•75   1716-53677 

146-86945   41-43110 

•25 

71868840 

9503317 

26-80836 

47-  1  1734-94454 

147-65485   41-65266 

•5 

730-61664 

95-81857 

27-02992 

•251  1753-45048 

148-44025!  4187422 

•75 

74264305 

96-603971  27-25147; 

•5 

1772-05460 

149-22565 

42-09577 

31- 

751-76763   97-38937;  27-47303  i   -75   1790-75639 

150-01104 

4231733 

•25 

766-99039"   98-17477 

27-69459!  48-  i  180955736 

150-79644 

42-53889 

•5 

779-31132 

9896016 

27-91614;   -25  ''  1828-45601 

151-58184 

42-76044 

•75 

791-73043 

99-74556 

28-13770 

•5    1847-45282 

152-36724 

42-98200 

32- 

804-24771 

100-53096 

2S-M3926;   -75   1866-54782   153-15264 

43-20356 

•25 

816-86317 

101-31636 

2M/59081I  49-'    1885-74099   153'93804 

4342511 

•5 

829-57631 

102-10176 

28-h0237;   -25!  1905-83233   154-72343 

43-64667 

•75 

842-38861 

102-887151  2902393 

•5   1924-42184 

155-50883   43-86823 

*33- 

855-29859 

10367255 

29-24548 

•75 

1943-90954 

156-29423   44-08978 

•25 

868-30675 

104-45795 

29-46704! 

50- 

1963-49540 

157-07963]  44-31134 

•5 

881-41308 

105-24335 

29-68860 

•25 

1983-17944 

157-96503]  44-53290 

•75 

89461759 

106-028751  29-91015 

•5 

2002-96166 

153-65042  44-75445 

34- 

907-92027   106-81415 

30-13171 

•75!  2022-84205 

159-43582!  44-97601 

25 

921-321131  107-59954 

30-35327 

51- 

2042-82062 

160-22122 

45-19757 

•5 

934-82016 

308-33494 

30-57482 

•25 

2062-89736 

161-00662 

45-41912 

•75 

94841736 

109-17034 

30-796'*8 

•5  !  2083-07227 

161-79202 

45-64068 

35- 

962-11275 

109-95574 

31-01794J  !   -751  210334536 

162-57741 

43-85224 

•25 

975-90630 

110-74114 

31-23949  52-  1  2123-71663 

163-36281 

46-08380 

•5 

989-79803 

111-52653!  31-46105 

•25|  2144-18607 

164-14821 

46-30535 

•75 

1003-78794 

112311931  31-68261 

•5  1  2164-75368 

184-93361 

46  52691 

36 

1017-87601 

11309733 

31-90416 

•75!  2185-41947 

165-71901 

46-74847 

•25 

1032-06227 

113-88-273 

32-12572 

53-    2206-18344 

166-50441 

46-97002 

•5 

•75 

1046-34670   11466813 
1060-729301  115-45353 

32-34728 

32-56383 

•25   2227-04557 
•5   2248-00589 

167-2-8980 
168-07520 

47-19158 
47-41314 

37- 

1075-21008 

116-23892 

32-7903J 

•75 

2269-06433 

168-86060 

47-63469 

•25 

1089-78903 

117-024  3-2 

33-01195 

54- 

2290-22104 

169-64600 

47-85625 

•5 

1104-46616 

117-80972 

33-23350 

•25:  2311-475^8 

170-43140 

48-07781 

•75 

1119-24141-   118-59572 

33-45506 

•5  !  2332-82889   171-21679  48-29936 

TABLE   OF  CIRCLES. 


65I 


Side  of 

&<leo/  ' 

Du.ni. 

Arciu 

Circuir,, 

equal  sq.       Diam.           Area. 

Circum. 

equal  ••!. 

"sflb 

2354-2800S      172-00219 

"~4~8l>2092 

71-5 

4015-1517f> 

~1^^23^!  ""63-36522 

55- 

2375-82344      172-78759 

48-74248 

•75 

404327833 

225-40927  i     63-58678 

•25 

2397-47698!     173-57299 

48-96403 

72- 

4071-50407 

226-19467 

63-80333 

•5        2419-22269      174  -3583  J 

49-18559 

•25 

4099-82750 

226-98006 

64-02989 

•75     2141-0665?!     175-14379 

49-40715 

•5 

4128-24909 

227-76546 

64-25145 

56- 

246300864!     175-92918 

49-62870 

•75 

4156-76886 

22855086 

64-47300 

•25 

2185-04887'     17<r7145Sl     49-85026 

73- 

4185-38681 

229-33626 

64-69456 

•5 

2507-18728      177-49998     5007182 

•25 

4214-10293 

230-12166 

64-91612 

•75      2520-452337  i     178-28538!     50-29337 

•5 

4242-91722 

230-90706 

65-13767 

57-         2551-75363I     179-07078      50-51493 

•75 

4271-82969 

231-69245 

65-35923 

•25|     2574-19156      179-85617 

50-73649 

74- 

4300-84034 

232-47785 

65-58079 

•5 

2596-72267      180-64157 

.  50-95804 

•25 

4329-94916 

23326325 

65-80234 

•75 

2619-35196      181-426b7|     51-17960 

•5 

4359-15615 

234-04865 

66-02390 

58- 

264207942!     18221237      51-40116 

•75 

4388-46132 

23483405 

66-24510 

•25|     26(54  90505;     182-99777 

51-62271 

75- 

4417-86466 

235-61944 

66-46701 

•5 

2687-82836      183-78317 

51  84427 

•25 

4447-36618 

236-40484 

66-6885? 

•75 

2710-85084    -  184-56856 

52-06583 

•5 

4476-965S8 

237-19024 

66-91043 

59- 

2733-97100      185-353  J6 

52-28738 

•75 

4506-66374 

237-97f)64 

67-13168 

•25 

275718933      18613936 

52-50894 

76- 

4536-45979 

238-76104 

67-3532  1 

•5 

2780-50584      186-92476 

52-73D50 

•25 

456635400 

239-54643 

67-57480 

•75 

280392053      187-71016 

52-95205 

•5 

45D6  34640 

240-33183 

67-79635 

60- 

2327-43338      188-49555      53-17364 

•75 

462643696 

241-11723 

6301791 

•25 

2851-04442      189-28095 

5339517 

77- 

465562571 

241-90263 

68-23J47 

•5 

2874-75362      193-06635 

5361672! 

•25 

468691262 

242-6H8U3 

68-46102 

•75 

2898-56101)      193-85175 

53-83328  ! 

•5 

471729771 

243-47343 

68-68258 

61 

2922-46656 

191-63715 

54-05H84! 

"75 

4747-78098 

24425382 

68-90414 

•25 

2946-47029 

192-42255 

5428139 

78- 

477836242 

245  04422 

63-12570 

•5 

2970-57220 

193-20794 

5451)295 

•25 

4809-04204 

24582962 

6931725 

•75 

2994-77228 

193-99334 

54-724511 

•5 

4839-81983 

24661502      69-56881 

62- 

3019-07054 

194-77874 

54<MG','6 

•75 

4870-79579 

247-40042      69-79037 

•25 

304346697 

195-56414 

55-167621 

79- 

4901-66993 

248-18591 

70-01192 

•5 

3J67-(J6I57 

196-34954 

55-33918 

•25 

4;>32-74225 

248-97121 

70-23318 

•75 

309255135 

197-13493 

5561073 

•5 

4963-91274 

24975661 

70-45504 

63 

3117-21531 

197-92033     55-83229 

•75 

4995-18140 

250-34201 

70-67659 

•25 

314203444 

198-70573      55-053-5 

80- 

5026-54824 

251-32741 

70-39815 

•5 

3166-92174 

199-49113      56-27510 

•25 

505801325 

252-11281 

71-11971 

•75 

3191-90722 

200-27653 

56-49698 

•5 

503957644: 

252-89820 

71-34126 

64- 

3*16-99087 

201-061^2 

5671852 

•75 

5  121  23781 

25368360 

71-56282 

•25 

3242-17270 

201-84732     56-9100? 

81- 

5152-99735 

254*46900 

71-78433 

•5 

326745270 

20263272      57-16163 

•25 

5184-85506 

255-25440 

72-00593 

•~5 

3292  83088 

203-41812      57-3  S3  19 

•5 

5216-SI<t'.l.~> 

!2^6-()3JSO 

72  -22749 

;35- 

3318-30784 

204-20352      57-60475 

•75 

5248-86501; 

25682579 

72-14905 

•25 

334388176 

204-98892     57-8263<)| 

82- 

5281-01725 

257-61059 

•  2-67060 

•5 

3369-55447 

20577431 

5804786' 

•25 

531326766 

258-3i)599 

72-89216 

•75 

339532534 

206-55971 

53-26942 

•5 

5345-r.ir.-ji 

259-18139 

7311372 

66- 

3421  19439 

207-34511 

5349097 

•75 

5378-06301 

259-96679 

?:<:«52? 

25 

344716162 

203-130511     53-712:-):! 

83- 

541060794 

260-75219 

?  3  55633 

•5 

3473  22702 

208-915911     58-93  lO'.i 

•25 

:»m-2f>K!:i 

261-53753 

73  77839 

•75 

349939060 

209-7013!)      5.H:V>!il 

•5 

r.l?.Vii9234 

262-32298 

73*99994 

67- 

3525-65235 

210  48670 

59-37720 

•75 

5508-83180 

263-10338 

74-22150 

25 

3552-01228 

211-27210 

595;»S76 

84- 

5.')  1  1  ?tV.t  1  1 

263-89378 

71:14306 

•5 

3573-47033 

212-05750 

59-821)31 

•25 

55?480f>2:> 

264-67918 

74-66461 

*T5 

3505  026i55 

212-84290      60-04  I*? 

•5 

5607-US923 

265-46457 

74-8861? 

68- 

3631-68110 

2i3-r,->>3o    r,o-2i;:m 

•75 

5641-17I3J 

266-24997 

75-10773 

•25 

9658-43373 

214-4  K«iy      CO-l-il'.tS 

85- 

."•67  1-50  173 

267-03537 

75'32:'-> 

•5 

S68528453 

21.")-  19.109      60-70C,;:! 

•25 

5707-y;{o-j:j 

2G7-82077 

75-55:  iM 

•75 

371!}  23350 

2J598449     60-92SIU 

•5 

5741  1  :,r,.  -j 

268-60617 

75-772  li  » 

69- 

3^39-28065 

216761IS9      C.l-ll.G.") 

•75 

5775-08178 

269-3915? 

75-99395 

•25 

'576  C  -42597 

217-f)5;-.29      61-37121 

86- 

:.s(is-H0481. 

270-176%      7621551 

•5 

379JJ-66947 

21834068 

61-59277 

•25 

5Sl262(i<)2 

270-96236      764370? 

•75 

332  '01  115 

21. 

61-81132 

•5 

5876-54540 

271-74770 

76-65362 

70 

:MS-|.-,KM)     vMH-'jiiis1    62-03588 

•75 

5.nor,r,29r. 

272-53316 

76-88018 

25 

3S7:>-<J3902j     2-20-tVjr.8S      62"J:>71I 

87- 

:.'J  1  1  67869 

27331856 

77-  10174 

•5 

3i)03-62522      2  i  1-43228      62-478'J9 

•25 

5978-892C.O 

274-10395 

77-:«:*2'.» 

•75 

3931-35959     MB-JJCTW     62-70055 

•5 

6013-20468 

274-88935 

77*54485 

71 

3959-19214'     •J-.'^or.WT      r>2'J2211 

•75 

604761  J91 

27567475 

7776641 

•2? 

3987-  12286      223  &i  i47      03  1  136l> 

88- 

6082-12337, 

276-16015 

77-98796 

652 


APPENDIX. 


Diam. 

Area. 

Circum. 

Side  of 
equal  sq. 

Dinm. 

Area. 

Circum. 

Side  ot 
equal  sq. 

88-25 

6116-729931  277-24555 

78-20952 

94-25 

6976-74097 

2-6-09510 

83-52688 

•5 

6151-43476  27803094 

78-43103 

•5 

7012-80194 

296-88050 

83-74844 

•75 

6186-23772  278-81634 

7865263 

•75 

7050-96109 

297-66590 

83-97000 

89- 

6-22M3385   279-60174 

78-87419 

95- 

7083-21842 

298-45130 

84-19155 

•35!  6256-13S15I  23033714 

79-09575 

•25 

7125-57992 

299-23070 

84-4131  1 

•5 

6291-23563 

231-17254 

79-31730 

•5 

7163-02759 

300-02209 

84-63467 

•75 

63-26-43129 

281-95794 

79-53386 

•75 

7200-57944 

300-80749 

84-85622 

90- 

6361-7-2512 

28-2-74333 

79-76042 

96- 

7233-22947 

3U  1-59239 

85-07778 

•25 

6397-11712 

233-52873 

79-98198 

•25 

7275-97767 

302-37829 

85-291)34 

•5 

6432-60730 

284-31413 

80-20353 

•5 

7313-82404 

303-16369   85-52039 

•75 

6463-19586 

285-09953 

80-42509 

•75)  7351-76859 

303-94908 

85-74245 

91- 

6503-83219 

285-83493 

80-64669 

97- 

7389-81131 

304-73448 

85-96401 

.25 

653J-66639 

286-67032 

80-85820 

•25 

7427-95221 

305-51988 

86-18556 

•5 

6575-54977 

237-45572 

81-08976 

•5 

7466-19129 

306-30523 

86-40712 

•75 

6611-53382 

288-24112 

81-31132 

•75 

7504-52853 

307-09068 

86-62868 

92- 

6647-61005 

239-02652 

81-53287 

98- 

7542-96396 

307-87608 

86-85023 

•25 

6683-78745 

289-31192 

81-75443 

•25 

7581-49755 

-  308-66147 

87-0717t' 

•5 

6720-06303 

2:)0-51J7:)'2 

81-975^'J 

•5 

7620-12933 

309-44637 

87-29335 

•75 

6756-43678 

291-33271 

82-VJ754 

•75 

7653-85927 

310-23227 

87-51490 

93- 

6792-90871 

292-16811 

82-41910 

99- 

7697-68739 

311-01767 

87-73646 

•25 

6829-47831 

292-95351 

82-04066 

•25 

7736-61369 

311-80307 

87-95802 

•5 

6866-14709 

293-7339} 

82-86-221 

•5 

7775-63816 

312-58346 

88-17957 

•75 

6902-91354 

294-52431 

83-08377 

•75 

7814-760311  313-37336 

88-40113 

94- 

6939-77317  295-30970,  83  30533  jj  100- 

7653-98163|  314-15926 

88-62269 

The  following  rules  are  for  extending  the  use  of  the  above  table. 

To  find  the  area,  circumference,  or  side  of  equal  square,  of  a  circle  having  a 
diameter  of  more  than  100  inches,  feet,  etc.  Rule. — Divide  the  given  diameter  by 
a  number  that  will  give  a  quotient  equal  to  some  one  of  the  diameters  in  the 
table  ;  then  the  circumference  or  side  of  equal  square,  opposite  that  diameter, 
multiplied  by  that  divisor,  or  the  area  opposite  that  diameter,  multiplied  by 
the  square  of  the  aforesaid  divisor,  will  give  the  answer. 

Example. — What  is  the  circumference  of  a  circle  whose  diameter  is  228 
feet  ?  228,  divided  by  3,  gives  76,  a  diameter  of  the  table,  the  circumference 
of  which  is  238-761,  therefore  : 

238-761 
3 


716 -283  feet.     Ans. 

Another  example. — What  is  the  area  of  a  circle  having  a  diameter  of  150 
inches?  150,  divided  by  10,  gives  15,  one  of  the  diameters  in  the  table,  the 
area  of  which  is  176-71458,  therefore: 

176-71458 

100  =  TO  X    IO 


17,671-45800  inches.     Ans. 

To  find  the  area,  circumference,  or  side  of  equal  square,  of  a  circle  having  an. 
intermediate  diameter  to  those  in  the  table.  Rule. — Multiply  the  given  diameter 
by  a  number  that  will  give  a  product  equal  to  some  one  of  the  diameters  in 
the  table  ;  then  the  circumference  or  side  of  equal  square  opposite  that  diame- 
ter, divided  by  that  multiplier,  or  the  area  opposite  that  diameter  divided  by 
the  square  of  the  aforesaid  multiplier,  will  give  the  answer. 


CAPACITY   OF   WELLS,   CISTERNS,    ETC.  653 

Example. — What  is  the  circumference  of  a  circle  whose  diameter  is  6J,  or 
6-125  inches?  6-125,  multiplied  by  2,  gives  12-25,  one  of  the  diameters  of  the 
table,  whose  circumference  is  38-484,  therefore  : 

2)38-484 
19-242  inches.     Ans. 

Another  example. — Wh?t  is  the  area  of  a  circle,  the  diameter  of  which  is  3-2 
feet?  3-2,  multiplied  by  5,  gives  16,  and  the  area  of  16  is  201-0619, therefore  : 

5  x  5  =  25)201-0619(8-0424  +  feet.     Ans. 
200 


106 

100 


61 
50 


Note.  —  The  diameter  of  a  circle,  multiplied  by  3-14159,  will  give  its  cir- 
cumference ;  the  square  of  the  diameter,  multiplied  by  -78539,  will  give  its 
area;  and  the  diameter,  multiplied  by  -88622,  will  give  the  side  of  a  square 
equal  to  the  area  of  the  circle. 

TABLE   SHOWING  THE  CAPACITY   OF  WELLS,    CISTERNS,    ETC. 

The  gallon  of  the  State  of  New  York,  by  an  act  passed  April  n,  1851,  is  required  to  conform  to 
the  standard  gallon  of  the  United  States  government.  This  standard  gallon  contains  231  cubic 
inches.  In  conformity  with  this  standard  the  following  table  has  been  computed. 

One  foot  in  depth  of  a  cistern  of 

3  feet  diameter  will  contain  ........................  52  •  872  gallons. 

3*  "  "  .......................   7I-965 

4  "  ........................   93-995         " 

4i  "  ........................  118-963 

5  "  "  ........................  146-868 

5i  "  "  ........................  177-710 

6  "  "  .....................  211-490        " 

6£  "  "  .......................  2*8-207 

7  "  "  .......................  287-861         " 

8  "  "  .......................  375-982 

9  .......................  475-852 

10  "  "          ............  :  ............  587-472 

12  "  "          .......................  845-959 

Note.  —  To  reduce  cubic  feet  to  gallons,  multiply  by  7-48.  The  weight  of  a 
gallon  of  water  is  8-355  Ibs.  To  find  the  contents  of  a  round  cistern,  multi- 
ply the  square  of  the  diameter  by  the  height,  both  in  feet,  and  this  product  by 

5-875- 


654 


APPENDIX. 


TABLE   OF   WEIGHTS. 


MATERIALS     USED    IN    THE    CONSTRUCTION    OR    LOADING    OF 

BUILDINGS. 

WEIGHTS  PER  CUBIC  FOOT. 

As  per  Barloiv,    Gallier,    Jlaswell,    Jfurst,    Rankine,    Tredgold,    Wood 
and  tJie  AutJwr. 


MATERIAL. 

| 
In 

£ 

AVERAGE. 

MATERIAL. 

S 

0 
K 

to 

o 
H 

AVERAGE. 

WOODS. 

Mahogany,  St.  Domingo.  .  .  . 

45 

65 
49 

55 
41 

41 

51 

46 

Mulberry  

35 

55 

45 

Alder 

35 

51 

38 

Oak,  Adriatic  

62 

49 

51 

5O 

"      Black   Bog  

60 

66 

63 

Ash                               

41 

57 

49 

"     Canadian  

54 

Beech 

46 

47 

Birch.  

35 

49 

42 

"     English  ..    

S 

7° 

54 

Box  . 

59 

65 

62 

Live  

57 

79 

68 

83 

"     Red  

47 

51 

64 

"     White. 

50 

Cedar 

27 

35 

31 

Olive..       

58 

47 

52 

44 

"     Palestine  
u     Virginia  Red         ...... 
Cherry  

30 
32 

.  38 
46 

34 
40 
39 

Pear-tree  .  .'  
Pine,  Georgia  (pitch)  
'*      Mar  Forest   

40 

38 

\ 

42 

48 
43 

Chestnut,  Horse 

29 

35 

"      Memel  and  Riga 

29 

** 

32 

Sweet  
Cork  .     ... 

27 

55 

41 
15 

"      Red  
"      Scotch     

27 

37 
39 

Cypress  . 

27 

34 

"      White 

28 

*'        Spanish  
Deal,  Christiania.    . 

40 
44 

"      Yellow  

27 
41 

39 

33 
45 

'*       English  
41       (Norway  Spruce). 

21 

t9 
7 

Poplar  

23 

37 

30 
44 

Dogwood  

47 

23 

Ebony 

69 

8^ 

76 

45 

Elder  

43 

30 

Elm  

31 

59 

46 

Satinwood  

e;5 

57 

27 

oA 

30 

"    (Red  Pine)  

3° 

37 

36 

38 

"    Riga 

47 

Teak 

61 

51 

53 

Tulip-tree 

30 

"    '  Water  

62 

Vine. 

81 

so 

37 

Walnut    Black 

•    26 

33 

Hemlock  

21 

26 

"         White'. 

eg 

49 

40 

la 

49 

Whitewood 

27 

te 

52 

Yew 

50 

Larch  .            ... 

31 

33 

"      Red 

31 

43 

METALS. 

"      White    

23 

Bismuth,  Cast 

614 

41 

83 

62 

487 

5O6 

41 

-?6 

544 

57 

k<       Plate 

q28 

531 

35 

38 

Bronze  . 

508 

516 

WEIGHT   OF   MATERIALS. 


655 


TABLE   OF    WEIGHTS.— (Continued.) 

MATERIALS    USED    IN    THE    CONSTRUCTION    OR    LOADING    OF 

BUILDINGS. 

WEIGHTS  PER  CUBIC  FOOT. 

As  per  Barlow,    Gallier,    Jfaswell,    Hurst,    Rankine,    Tredgold,    Wood 
and  the  Author. 


MATERIAL. 

Z 

1 

o 
H 

AVERAGE. 

MATERIAL. 

I 

t* 

O 

H 

AVERAGE. 

Copper    Cast 

537 

549 

543 

06 

u_ 

1O4 

556 

dry  

1OO 

"         Plate           

644 

44         in  Cenfent  . 

112 

Gold 

1206 

"         in  Mortar 

1  1O 

110S 

Caen  Stone 

13O 

509 

81 

475 

487 

481 

"         Roman,  Cast. 

1001 

Cast 

434 

474 

454 

"       Malleable  
"       Wrought 

486 

475 

480 

equal  parts.. 
Chalk 

116 

113 

14  * 

1  Lead    Cast 

709 

Clay                   

122 

Er^lish  Cast 

717 

"     with  Gravel 

16O 

"       M  ficd 

713 

90 

851 

?6 

102 

83 

"          "     60° 

849 

8i 

79 

"               "    212° 

837 

85 

Nickel   Cast 

488 

Coke 

^.6 

62 

54  I 

453 

Concrete,  Cement  

13O 

975 

1O6 

1345 

U  o 

n            •'     Rolled 

1379 

126 

142 

"        with  Gravel  

126 

636 

25O 

*'    '  Pure  Cast 

655 

Feldspar 

16O 

"           "         "     Hammered 
"       Standard 

... 

658 
644 

Flagging,  Silver  Gray.    .   .  . 
Flint                  

-  - 

185 
163 

Steel 

<86 

402 

489 

l6e 

160 

Tin   Cast 

!,6 

1  •< 

462 

"       Flint 

183 

429 

449 

439 

"       Green 

l«i;» 

Zinc,  Cast  

'«       Plate          

163 

"      White 

167 

181 

174 

STONES,  EARTHS,  ETC. 

Granite 

nR 

Hi  5 

161 

x6c 

180 

173 

166 

156 

"       Guernsey  

185 

8O 

166 

277 

Gnvel          

9° 

12O 

105 

Basalt 

187 

134 

J!-5 

i  •{  M 

135 

1^5 

14O 

t)atn    i  one..^.. 

129 

Lime    Unslaked         

52 

r>f        c              r'    

16O 

11Q 

169 

Brick           ' 

85 

102 

Aubigne  

146 

138 

162 

,i      -»T    TJ  h  '  °d 

107 

Marble                      

161 

178 

17O 

"        "      Salmon 

' 

1OO 

166 

105 

17O 

656 


APPENDIX. 


TABLE   OF   WEIGHTS.— (Continued^ 

MATERIALS    USED     IN    THE     CONSTRUCTION    OR    LOADING    OF 

BUILDINGS. 

WEIGHTS  PER  CUBIC  FOOT. 

As  per   BarloTV,    Gallier,    Haswell,    Hurst,  Rankine>    Tredgold,    Wood 
and  the  Author. 


MATERIAL. 

1 

S 

AVERAGE. 

MATERIAL. 

6 

N 

CK 

o 
H 

AVERAGE. 

Marble    Enstchester 

167 
i63 

IOO 
110 

178 
169 

179 
140 

173 
167 
166 
167 
140 
125 
175 
155 
98 
103 
107 

9 
86 
105 
1OO 

118 
83 
146 
72 
8O 
180 
175 
147 
56 
165 
165 
124 
112 
1O5 
105 
123 
1O5 
97 
172 
126 
144 
133 
142 
134 
141 
134 
162 
150 

Serpentine  
"         Chester    Pa 

... 

165 
144 
152 
95 
159 
167 
157 
18O 
135 
151 
14O 
16O 
14O 
124 
115 
170 
76 

SS 
49 
59 
62 
26 
2O 
58 
57 
61 
17 
61 
69 
114 
73 
131 
559 
68 
171 
133 
131 
14 
8O 
62} 
64 
81 

Egyptian  
"          French  

Shingle  

137 

181 

Marl                   

Slate  

Mica 

"     Cornwall  

109 

118 

"     Welsh           

87 

83 

Stone,   Artificial  

120 

150 

"         Paving     .   ,.„  

Stone-work  .    

120 

i€o 

"       Hair,  incl.  Lath  and 
Nails,  per  foot  sup. 

7 

ii 

"           Rubble  
Sulphur,  Melted  

"    Sand  3  and  Lime  paste  2 
"        "     3    "        "         "      2 
well  beat  together.  . 
Peat    Hard       

Trap  Rock 

MISCELLANEOUS. 

Petrified  Wood       

Pitch 

Plaster   Cast  

Bark,  Peruvian  

"        '  Red 

132 

161 

17 

»4 

34 
25 

62 

24 
66 

Charcoal 

Pumice-stone  

Cotton,  baled.  .  .   
Fat                 

52 

JO 

56 

Rotten-stone  
Sand,  Coarse    

92 
90 
118 

ii8 

120 
128 

Gutta-percha  
Hay   baled                 

India  Rubber 

'      Dry 

Moist 

Pit 

92 

101 

Quartz 

Red  Lead 

158 

130 

Rock  Crystal 

"          Amherst,  O.  .     .  . 
"          Belleville,  N.  J... 
Berea,  O  
"          Dorchester,  N.  S 
"     •«          Little  Falls,  N.  J. 
u          Marietta,  O.   .    .  . 
"          Middletown,  Ct.. 

Salt 

20 
100 

Saltpetre  —  
Snow  

"a 

60 

Water,  Ram   
"       Sea                   

Whalebone 

INDEX. 


Abscissas  of  Axes,  Ellipse 484 

Abutments,  Bridges,.  Strength  of.  227 
Abutments,  Houses,  Strength  of..  53 

Acute  Angle  Denned 349,  544 

Acute-angled  Triangle  Denned...    545 

Acute  or  Lancet  Arch 51 

Algebra,  Addition 398 

Algebra,  Application  of 393 

Algebra,     Binomial,     Multiplica- 
tion of ,  409 

Algebra,  Binomial,  Square  of  a...  429 
Algebra,  Binomial,  Squaring  a. . .  410 

Algebra  Defined 392 

Algebra,      Denominator,       Least 

Common 404 

Algebra,  Division,  the  Quotient..  419 
Algebra,  Division,  Reduction....  419 
Algebra,  Division,  Reverse  of 

Multiplication 418 

Algebra,    Factors,    Multiplication 

of  Two  and  Three 409 

Algebra,    Factors,    Multiplication 

of  Three . 408 

Algebra,  Factors,  Squaring  Differ- 
ence of  Two 412 

Algebra,     Fractions    Added    and 

Subtracted 403 

Algebra,  Fractions,  Denominators  407 
Algebra,  Fractions  Subtracted...  405 
Algebra,  Hypothenuse,  Equality 

of  Squares  on 416 

Algebra,  Letters,  Customary  Uses 

of 396 

Algebra,  Logarithms  Explained..  425 
Algebra,  Logarithms,  Examples  in  426 
Algebra,  Multiplication,  Graphical  408 


Algebra,  Progression,  Arithmeti- 
cal...   432 

Algebra,  Progression,  Geometrical  435 

Algebra,  Proportion  Essential 347 

Algebra,      Proportionals,      Lever 

Formula 421 

Algebra,  Quantities,  Addition  and 

Subtraction 424 

Algebra,  Quantities,  Division  of.   424 
Algebra,    Quantities,    Multiplica- 
tion of 424 

Algebra,  Quantities  with  Negative 

Exponents 423 

Algebra,  Quantity,  Raising  to  any 

Power 423 

Algebra,  Radicals,  Extraction  of..  425 

Algebra,  Rules  are  General 394 

Algebra,  Rules,  Useful  Construc- 
ting   394 

Algebra,  Signs 397 

Algebra,  Signs,  Arithmetical  Pro- 
cess by 396 

Algebra,  'Signs,    Changed    when 

Subtracted ....  400 

Algebra,  Signs,  Multiplication  of 

Plus  and  Minus. .. 415 

Algebra,  Squares  on  Right- Angled 

Triangle 417 

Algebra,  Subtraction 398 

Algebra,  Sum  and  Difference,  Pro- 
duct of 413 

Algebra,  Symbols  Chosen  at  Pleas- 
ure    395 

Algebra,  Symbol,  Transferring  a..   399 
Algebra,    Triangle,     Squares     on 
Right-angled 417 


658 


INDEX. 


Alhambra,  or  Red  House,  Ancient 

Palace  of  the n 

Ancient    Cities,     Historical     Ac- 
counts of 6 

Ancient  Monuments,  their  Archi- 
tects        6 

Angle  at  Circumference  of  Circle.  358 

Angle  Defined 544 

Angle   to  Bracket  of  Cornice,  To 

Obtain 343 

Angle,  To  Measure  a,  Geometry..  348 
Angle  rib  to  Polygonal  Dome. . . .  223 
Angle-rib,  Shape  of  Polygonal 

Domes 223 

Amulet  or  Fillet,  Classic  Mould- 
ing  323 

Antae  Cap,  Modern  Moulding.  ...   334 

Antique  Columns,  Forms  of 48 

Antiquity  of  Building 5 

Arabian  and  Moorish  Styles,  An- 
tiquities of ii 

Araeostyle,  Intercolumniations. . .     20 

Arc  of  Circle  Defined 547 

Arc  of  Circle,  Length,  Rule  for. .  475 

Arc,  Radius  of,  To  Find 561 

Arc,  Versed  Sine,  To  Find  (Geom- 
etry)   561 

Arcade 52 

Arcade  of  Arches,  Resistance  in. .  52 
Arcade  in  Bridges,  Strength  of 

Piers 52 

Arch 50 

Arch,  Acute  or  Lancet 51 

Arch,  Archivolt  in 52 

Arch,  Bridge,  Pressure  on 51 

Arch,  Building,  Manner  of 50 

Arch,  Catenary 51 

Arch,  Construction  of 50 

Arch,  Definitions  and  Principles  of    52 

Arch,  Extrados  of 52 

Arch,  Form  of 50 

Arch,  Formation  in  Bridges 51 

Arch,  Hooke's  Theory  of  an 50 

Arch,  Horseshoe  or  Moorish 51 

Arch,  Impost  in 52 

Arch,  Intrados  of 52 

Arch,  Keystone,  Position  of 50 

Arch,  Lateral  Thrust  in .     52 


Arch,  Ogee 51 

Arch,  Rampant 51 

Arch,  Span  of  an 52 

Arch,  Spring  in  an 52 

Arch,  Stone  Bridges 230 

Arch-stones,  Bridges,  Jointing. ..  233 

Arch,  Strength  of 50 

Arch  of  Titus,  Composite  Order. .  28 

Arch,  Uses  of 50 

Arch,  Voussoir  in 52 

Architect  and   Builder,  Construc- 
tion Necessary  to 56 

Architect,  Derivation  of  the  Word  5 

Architects  of  Italy,  I4th  Century.  12 
Architecture,   Classic    Mouldings 

in 323 

Architecture,  Ecclesiastical,  Origin 

of 14 

Architecture,  Egyptian,  Character 

of 33 

Architecture,    Egyptian,  Features 

of 30 

Architecture,     English,      Ccttage 

Style 35 

Architecture,  English,  Early n 

Architecture,  Grecian  and  Roman  8 

Architecture,  Grecian,  History  of.  6 
Architecture,    Hindoo,   Character 

of . 30 

Architecture,  Order,  Three  Princi- 
pal parts  of 14 

Architecture,  Principles  of 44 

Architecture,  Roman,  Ruins  of...  II 

Architecture  in  Rome  Defined. ..  7 

Architecture,  Result  of  Necessity.  13 

Architrave  Defined 15 

Area  of  Circle,  To  Find 475 

Area  of  Post,  Rule  for  Finding. . .  90 

Area  of  Round  Post,  Rule 90 

Area  of  Surface,  Sliding  Rupture, 

Rule 88 

Arithmetical    Progression    (Alge- 
bra)   432 

Astragal,  or  Bead,  Classic  Mould- 
ing   323 

Athens,  Parthenon,  Columns  of. .  48 
Attic,    a     Small    Order,    Top    of 

Building 15 


INDEX. 


659 


Attic  Story,  Upper  Story 15 

Axes  of  Ellipse  (Geometry) . .  585 

Axiom  Defined  (Geometry) 348 

Axis  Defined 548 

Balusters,  Handrailing,  Winding 
Stairs , 310 

Baluster,  Platform  Stairs,  Position 
of 250 

Baluster  in  Round  Rail,  Winding 
Stairs 313 

Base,  Shaft,  and  Capital  Defined .     14 

Bathing,  Necessary  Arrangements 

for 45 

Baths  of  Diocletian,  Splendor  of..     27 
Bead,  or  Astragal,  Classic  Mould- 
ings   323 

Beams,    Bearings    of,    Rules    for 

Pressure 75 

Beams,  Breaking  Weight  on 74 

Beams,  Framed,  Rules  for  Thick- 
ness   130 

Beams,  Framed,  Position  of  Mor- 
tise   236 

Beams,  Headers  Defined 130 

Beams,  Horizontal   Thrust,  Rule.s 

for 72 

Beams,  Inclined,  Effect  of  Weight 

on 72 

Beams,  Load  on,  Effect  of 74 

Beams,  Splicing 235 

Beams,  Tail,  Defined 130 

Beams,  Trimmers  or  Carriage,  De- 
fined    130 

Beams,  Weight  on,  Proportion  of.   130 
Beams,    White    Pine,    Table     of 

Weights 177 

Beams,  Wooden,  Use  of  Limited.  154 

Bearings  for  Girders 141 

Binomials,  Multiplication  of  (Al- 
gebra)   409 

Binomials,  Square  of  (Algebra). . .  429 
Binomials,  Squaring  (Algebra)...   410 

Bisect  an  Angle  (Geometry) 554 

Bis'ect  a  Line  (Geometry) 549 

Blocking  out  Rail,  Winding  Stairs  301 
Blondel's  Method,  Rise  and  Tread 
in  Stairs 242 


Bottom   Rail  for  Doors,  Rule  for 

Width 316 

Bow,    Mr.,    On    Economics    and 

Construction 166 

Bowstring      Girder,      Cast -Iron, 

Should  not  be  Used 163 

Brace,  Length  of,  To  Find  (Geom- 
etry)    579 

Braces,    Rafters,    etc.,    To    Find 

Length 580 

Braces  in  Roof,  Rule  for,  Same  as 

Rafter 208 

Breaking  Weight  Defined 84 

Brick  or  Stone  Buildings 37 

Brick  Walls,  Modern 49 

Bridge  Abutments,  Strength  of.. .   227 

Bridge  Arches,  Formation  of 51 

Bridge  Arch-stones,  Joints  of ....  233 
Bridges,  Construction  of  Various.  223 
Bridge,  London,  Age  of  Piles 

under 229 

Bridge    Piers,    Construction   and 

Sizes 228* 

Bridge,  Rib-built 224 

Bridge,  Rib,  Construction  of. . .  .  225 
Bridge,  Rib,  Framed, Construction 

and  Distance 226 

Bridge,  Rib,  Radials  of 226 

Bridge,  Rib,  Table   of  Least  Rise 

in 224 

Bridge,  Rib,  Rule  for  Area  of 225 

Bridge,  Rib,  Rule  for  Depth  of. . .  226 

Bridge,  Roadway,  Width  of 227 

Bridge,  Stone,  Arch  Construction  230 
Bridge,  Stone,  Arch-stones,  Table 

of  Pressures  on 230 

Bridge,  Stone,  Arch,  Centres  for, 

Bad  Construction 229 

Bridge,  Arch,  Spring  of 247 

Bridge,  Stone,  Strength  of  Truss- 
ing    232 

Bridge,  Weight,  Greatest  on! 225 

Bridge,  without  Tie-Beam 224 

Bridging,       Cross-,      Additional 

Strength  by 137 

Bridging,  Cross-,  Defined 137 

Bridging,    Cross-,    Resistance   by 
Adjoining  Beams 139 


66o 


INDEX. 


Building,  Antiquity  of 5 

Building,  Elementary  Parts  of  a..  46 

Building,  Expression  in  a 35 

Building  by  the  Greeks 35 

Building,  Modes  of,  Defined 9 

Building  by  the  Romans 26 

Building,  Style  of,  Selected  to  Suit 

Destination 35 

Butt-joint  on  Handrail  to  Stairs. .  303 
Butt-joint,  Handrail,  Stairs,  Posi- 
tion of 307 

Byzantine  Style,  Lombard 10 

Campanile,   or    Leaning    Tower, 

Twelfth  Century 12 

Capital,     Uppermost    Part    of    a 

Column 15 

Carriage  Beam,  Well-Hole  in  Mid- 
dle, Find  Breadth 136 

Carriage  Beam,  One  Header,  Rule 

for  Breadth 133 

Carriage   Beam   or  Trimmer  De- 
fined   rso 

Carriage  Beam,  Rule  for  Breadth.  132 
Carriage  Beam,  Two  Sets  of  Tail 

Beams,  Rule  for  Breadth 134 

Caryatides,   Description  and  Ori- 

.  gin  of 26 

Cast-Iron       Bowstring       Girder, 

Should  not  be  Used 163 

Cast-Iron  Girder,  Load  at  Middle, 

Size  of  Flanges. 162 

Cast-Iron  Girder,  Load    Uniform, 

Size  of  Flanges 163 

Cast-iron  Girder,  Manner  of  Mak- 
ing a 161 

Cast-Iron  Girder,  Proper  Form...  161 

Cast-iron,  Tensile  Strength  of. ...  161 

Cast-Iron  Untrustworthy 161 

Catenary  Arch,  Hooke's  Theory  of.  51 

Cathedral  of  Cologne n 

Cathedrals,  Domes  of 53 

Cathedrals   of   Pisa,    Erection   in 

1016 12 

Cavern,    The   Original    Place    of 

Shelter 13 

Cavetto  or  Cove,  Classic  Moulding  323 

Cavetto,  Grecian  Moulding 327 


Cavetto,  Roman  Moulding 329 

Ceiling,  Cracking,   How   to    Pre- 
vent    125 

Centre  of  Circles,  To  Find  (Ge- 
ometry)....  556 

Centre  of  Gravity,  Position  of 71 

Centre  of  Gravity,  Rule  for  Find- 
ing, Examp^es 71 

Chimneys,  How  Arranged, 42 

Chinese  Structure,  The  Tent  the 

Model  of . .  *. 14 

Chord  of  Circle  Defined 547 

Chords  Giving  Equal  Rectangles.  363 
Circle,  Arc,  Rule  for  Length  of. . .  475 
Circle,  Area,  Circumference,  etc., 

Examples 652 

Circle,  Area,  Rule  for,  Length  of 

Arc  Given 478 

Circle,  Area,  To  Find 475 

Circle>  Circumference,  To  Find  . .  473 

Circle  Defined 546 

Circle,  Describe  within  Triangle..   566 
Circle,  Diameter  and   Circumfer- 
ence   472 

Circle,  Diameter  and  Perpendicu- 
lar  468 

Circle   Equal    Given    Circles,   To 

Make 580 

Circle,  Ordinates,  Rule  for 471 

Circle,    Radius   from    Chord   and 

Versed  Sine 469 

Circle,  Sector,  Area  of 476 

Circle,  Segment,  Area  of 477 

Circle,  Segment  from  Ordinates. .  470 
Circle,  Segment,  Rule  for  Area  of.  479 

Circles,  Table  of 649-652 

Circle  through  Given  Points 559 

Circular  Headed  Doors 320 

Circular  Headed  Doors,  To  Form 

Soffit 321 

Circular  Headed  Windows 320 

Circular    Headed    Windows,    To 

Form  Soffit 321 

Circular  Stairs,  Face  Mould  for  (i).  282 
Circular  Stairs,  Face  Mould  for  (2).  285 
Circular  Stairs,  Face  Mould  for  (3). -287 
Circular  Stairs,  Face  Mould,  First 
Section 283 


INDEX. 


66l 


Circular  Stairs,  Falling  Mould  for 

Rail 281 

Circular  Stairs,  Handrailing  for  . .  278 

Circular  Stairs,  Plan  of 279 

Circular  Stairs,  Plumb  Bevel  De- 
fined    282 

Circular   Stairs,    Timbers    Put   in 

after  Erection *. . .  253 

Cisterns,    Wells,    etc.,    Table    of 

Capacity  of 653 

City  Houses,  General  Idea  of.. ...     42 
City  Houses,  Arrangements  for.. .     37 

Civil  Architecture  Defined 5 

Classic   Architecture,    Mouldings 

in 323 

Classic  Moulding,  Annulet  or  Fil- 
let   323 

Classic     Moulding,    Astragal     or 

Bead 323 

Classic      Moulding,    Cavetto     or 

Cove 323 

Classic  Moulding,  Cyma-Recta. . .  324 

Classic  Moulding,  Cyma-Reversa.  324 

Classic  Moulding,  Ogee 324 

Classic  Moulding,  Ovolo 323 

Classic  Moulding,  Scotia 323 

Classic  Moulding,  Torus 323 

Coffer  Walls 49 

Cohesive  Strength  of  Materials. . .  76 

Collar  Beam  in  Truss 238 

Cologne,  Cathedral  of u 

Columns,  Antique,  Form  of 48 

Column,  Base,  Shaft  and  Capital.  14 
Columns,  Egyptian,  Dimensions, 

etc 33 

Column,  Gothic  Pillar,  Form  of..  48 

Column,  Outline  of. 47 

Columns,   Parthenon    at   Athens, 

Forms  of 48 

Column  or  Pillar 47 

Column,  Resistance  of 47 

Column,  Shaft,  Form  of. 47 

Column,   Shaft,   Swell   of,  Called 

Entasis 48 

Complex,  or  Ground  Vault 52 

Composite  Arch  of  Titus 28 

Composite,  Corinthian  or  Roman 

Order..                    28 


Compression,  Resistance  to 77 

Compression,  Resistance  to  Crush- 
ing and  Bending.*. 85 

Compression,  Resistance  to,  Pres- 
sures Classified 83 

Compression,  Resistance  to,  Table 

of 79 

Compression,     Resistance    to,    in 

Proportion  to  Depth 101 

Compression  at  Right  Angles  and 

Parallel  to  Length 206 

Compression  of  Stout  Posts 89 

Compression        and        Tension, 

Framed  Girders 174 

Compression  Transversely  to  Fi- 
bres   86 

Cone  Defined 548 

Conic  Sections 584 

Conjugate  Axis  Defined 548 

Conjugate  Diameters  to  Axes  of 

Ellipse 487 

Construction  Essential 56 

Construction     of    Floors,    'Roof, 

etc.,  Economy  Important 123 

Construction,     Framing,     Heavy 

Weight 56 

Construction,    Joints,    Effect     of 

Many 123 

Construction,  Object  of  Defined. .   123 
Construction,  Simplest  Form  Best.  123 
Construction,   Superfluous   Mate- 
rial      56 

ontents,  Table  of,  General. .  .613-624 
orinthian  Capital,  Fanciful   Ori- 
gin of. 24 

orinthian  Order  Appropriate  in 

Buildings.. . . . 24 

orinthian  Order,  Character  of....  16 
orinthian  Order,  Description  of.  23 
orinthian  Order,  Elegance  of. ...  23 
orinthian  Order,  The  Favorite 

at  Rome 27 

orinthian  Order,  Grecian  Origin 

of 16 

orinthian  Order,  Modification  of.     27 
Cornice,  Angle  Bracket,  To  Ob- 
tain the 343 

Cornice,  Eaves,  To  Find  Depth  of.  335 


662 


INDEX. 


Cornice,  Mouldings,  Depth  of. . . .  342 
Cornice,  Projection,  To  Find  ....  342 
Cornice,   Projecting  Part  of  En- 
tablature       15 

Cornice,  Rake  and  Level  Mould- 
ings, To  Match 344 

Cornice,  Shading,  Rule  for .   6n 

Cornice,  Stucco,  for  Interior,  De- 
signs    340 

Corollary  Defined  (Geometry) 348 

Corollary  of  Triangle  and  Right 

Angle 355 

Cottage  Style,  English 35 

•  Country-Seat,  Style  of  a 37 

Cross-Bridging,      Additional 

Strength  by 137 

Cross-Bridging,    Furring     Impor- 
tant. .;..• 137 

Cross-Bridging,  Resistance  of  Ad- 
joining Beams  ... 139 

Cross-  or  Herring-Bone  Bridging 

Defined 137 

Cross-Furring  Denned 125 

Cross-Strains,  Resistance  to. . .  .77,  99 
Crushing  and  Bending  Pressure..  85 
Crushing,  Liability  of  Rafter  to. . .  205 
Crushing  Strength  of  Stout  Posts.  89 

Cube  Root,  Examples  in 645 

Cubes,  Squares  and  Roots,  Table 

of 638-645 

Cubic  Feet  to  Gallons,  To  Reduce.  653 

Cupola  or  Dome 53 

Curb  or  Mansard  Roof 54 

Curve  Ellipse,  Equations  to 482 

Curve  Equilibrium  of  Dome 218 

Cylinder,  Defined 549 

Cylinder,  Platform  Stairs 248 

Cylinder,  Platform  Stairs,  Lower 

Edge  of 249 

Cylinders  and  Prisms,  Stair-Build- 
ing    257 

Cyma-Recta,  Classic  Moulding...  324 
Cyma-Reversa,  Classic  Moulding.  324 
Cyma-Recta,  Grecian  Moulding  . .  327 
Cyma-Reversa,  Grecian  Moulding.  328 

Deafening,  Weight  per  Foot 177 

Decagon,  Defined 546 


Decimals,  Reduction  of,  Examples  647 

Decorated  Style,  i4th  Century. ...     u 

Decoration,  Attention  to  be  given 
to.. .: 46 

Decoration,  Roman 27 

Deflection,  Defined 112 

Deflection,  Differs  in  Different  Ma- 
terials    113 

Deflection,   Elasticity  not  Dimin- 
ished by 112 

Deflection,    Floor-beams,    Dwell- 
ings, Dimensions 127 

Deflection,     Floor-beams,    First- 
class  Stores,  Dimensions 128 

Deflection,  Floor-beams,  Ordinary 
Stores,  Dimensions 127 

Deflection,  Lever,  Principle  of. .  .    119 

Deflection,  Lever  and  Beam,  Rela- 
tion Between 119 

Deflection,   Lever,  To   Find,  Load 
at  End 120 

Deflection,      Lever,      Breadth     or 
Depth,  Load  at  End 121 

Deflection,  Lever,  Load  Uniform.   121 

Deflection,      Lever,     Breadth     or 
Depth,  Load  Uniform 122 

Deflection,     Lever,     for     Certain, 
Load  Uniform 122 

Deflection,    Load  Uniform    or   at 
Middle,  Proportion  of 116 

Deflection,  Load  Uniform,  Breadth 
and  Depth 117 

Deflection,    Load    Uniform    or   at 
Middle,  Proportion  of 119 

Deflection,      in      Proportion      to 
Weight '. .   112 

Deflection,    Resistance    to,    Rule 
for 113 

Deflection,  Safe  Weight   for    Pre- 
vention..     no 

Deflection,    Weight    at     Middle, 
Breadth  and  Depth 114 

Deflection,  Weight  at  Middle,  for 
Certain 114 

Deflection,  Weight  at  Middle,  Cer- 
tain, for 116 

Deflection,  Weight    Uniform,    for 
Certain , 117 


INDEX. 


Deflection,  Weight  Uniform,  Cer- 
tain, for 

Denominator,  Least  Common  (Al- 
gebra)  '. 

Denominator,  Least  Common 
(Fractions) 

Dentils,  Teeth-like  Mouldings  in 
Cornice 

Diagonal  Crossing  Parallelogram 
(Geometry) 

Diagonal  of  Square  Forming  Oc- 
tagon   

Diagram  of  Forces,  Example.... 

Diameter,   Circle,  Denned 

Diameter,  Ellipse,  Denned 

Diastyle,  Explanation  of  the  Word 

Diastyle,  Intercolumniation 

Diocletian,  Baths,  Splendor  of. ... 

Division,  Fractions.  Rule  for 

Division,  by  Factors  (Fractions).. 

Division,  Quotient  (Algebra) 

Division,  Reduction  (Algebra)... 

Dodecagon,  Denned 

Dodecagon,  To  Inscribe 

Dodecagon,  Radius  of  Circles 
(Polygons) 

Dodecagon,  Side  and  Area  (Poly- 
gons)  

Dome,  Abutments,  Strength  of. . . 

Domes  of  Cathedrals 

Dome,  Character  of '. 

Dome,    Construction    and    Form 


118 


404 


384 


357 
1 66 

547 

549 

19 

20 

27 

389 

381 

419 


Dome,  Small,  over  Stairways,  Form 

of 220 

Dome,  Spherical,  To  Form 221 

Dome,  St.  Paul's,  London 54 

Dome,  Strains  on,  Tendencies  of.   219 

Domes,  Wooden 54 

Doors,  Circular  Head 320 

Doors,  Circular   Head,    to    Form 

Soffit 321 

Doors,  Construction  of 317 

Doors,  Folding  and  Sliding,  Pro- 
portions   316 

Doors,  Front,  Location  of 320 

Doors,    Height,    Rule   for,  Width 

Given 315 

Door  Hanging,  Manner  of 317 

Doors,  Panel,   Bottom  and  Lock 

Rail,  Width 316 

Doors,  Panel,  Four  Necessary. . .  317 
Doors,  Panel,  Mouldings,  Width.  317 
Doors,  Panel,  Styles  and  Muntins, 


419       Width 316 

546  Doors,  Panel,  Top  Rail,  Width. ..  317 
569  Doors,  Stop  for.  How  to  Form...  317 
Doors,  Single  and  Double,  height 

452  of 316 

Doors,  Trimmings  Explained. ...   317 

453  I  Doors,  Uses  and  Requirements  of  315 

Doors,  Width  of 315 


of. 


216 


Dome,  Construction  and  Strength 
of 

Dome,  Cubic  Parabola  computed 

Dome  or  Cupola,  the 

Dome,  Curve  of  Equilibrium,  rule 
for 

Dome,  Halle  du  Bled,  Paris 

Dome,  Pantheon  at  Rome 

Dome,  Pendentives  of 

Dome,  Polygonal,  Shape  of  An- 
gle Rib 

Dome,  Ribbed,  Form  and  Con- 
struction  

Dome,  Scantling  for,  Table  of 
Thickness 


53 
219 

53 


218 

54 
53 
53 


223 


217 


218 


Doors,  Should  not  be  Winding. ..  317 
Doors,  Width  and  Height,  Propor- 
tion of 315 

Doors,  Width,    Rule   for,    Height 

Given 316 

Doric  Order,  Character  of 16 

Doric  Order,  Grecian  Origin  of. .     16 
Doric  Order,  Modified  by  the  Ro- 
mans      27 

Doric  Order,  Used  by  Greeks  only 

at  First 19 

Doric  Order.  Peculiarities  of 17 

Doric  Order,  Rudeness  of 30 

Doric  Order,  Specimen  Buildings 

in 19 

Doric   Temples,   Fanciful    Origin 

of 17 

Doric  Temples 19 

Drawing,  Articles  Required 536 


664 


INDEX. 


Drawing-board      Better     without 

Clamps 537 

Drawing-board    Liable    to   Warp, 

How  Remedied 537 

Drawing-board,        Difficulty       in 

Stretching  Paper 539 

Drawing-board,  Ordinary  Size 536 

Drawing,    Diagrams    aid    Under- 
standing   536 

Drawing,  Inking  in 542 

Drawing,  Laying  Out  the 541 

Drawing,  the  Paper 537 

Drawing  in  Pencil,  To  Make  Lines  542 

Drawing,  Secure  Paper  to  Board.  537 

Drawing,  Shade  Lining 543 

Drawing,  Stretching  Paper 537 

Durability  in  a  Building 37 

Dwelling,  Arrangement  of  Rooms  38 
Dwellings,  Floor-beams,  To  Find 

Dimensions  127 

Dwellings,      Floor-beams,       Safe 

Weight  for 126 

Dwelling-houses,  Dimensions  and 

Style •. 37 

Eaves  Cornice,  Designs  for 335 

Eaves  Cornice,  Rule  for  Depth...   335 
Ecclesiastical  Architecture,  Point- 
ed Style ii 

Ecclesiastical  Style,  Origin  of. ...     14 

Echinus,  Grecian  Moulding 327 

Economy,    Construction    Floors, 

Roofs,  Bridges 123 

Eddystone  and   Bell  Rock    Light 

House 48 

Egyptian  Architecture 30 

Egyptian  Architecture,  Appropri- 
ate Buildings  for 33 

Egyptian  Architecture,  Character 


of. 


33 

Egyptian  Architecture,  Origin  in 

Caverns 14 

Egyptian  Architecture.  Principal 

Features  of 30 

Egyptian  Columns,  Dimensions 

and  Proportions 33 

Egyptian  Walls,  Massiveness  of.  .  33 

Egyptian  Works  of  Art  30 


Elasticity  of  Materials 84 

Elasticity  not   Diminished  by  De- 
flection     112 

Elasticity,   Result    of    Exceeding 

Limit 120 

Elevation,  a  Front  View 37 

Elevated    Tie-beam    Roof    Truss 

Objectionable 214 

Ellipse,  Area 488 

Ellipse,  Axes,  Two,  To  Find,   Di- 
ameter and  Conjugate  Given. ..  593 

Ellipse  Defined 481 

Ellipse,  Equations  to  the  Curve. .  482 
Ellipse,  Major   and    Minor   Axes 

Defined : ..  481 

Ellipse,  Ordinates,  Length  of  ...  491 
Ellipse,  Parameter  and  Axis,   Re- 
lation of 485 

Ellipse,  Practical  Suggestions....  489 
Ellipse,  Semi-major,  Axis  Defined  486 

Ellipse,  Subtangent  Defined 486 

"Ellipse,  Tangent   to    Axes,  Rela- 
tion of 485 

Ellipse,  Tangent  with  Foci,  Rela- 
tion of 487 

Ellipsis,  Axes  of,  To  Find  (Geom- 
etry)    585 

Ellipsis, Conjugate  Diameters  (Ge- 
ometry)    593 

Ellipsis  Defined 548,  585 

Ellipsis,  Diameter  Defined 549 

Ellipsis,  Foci,  To  Find 586 

Ellipsis,  by  Intersecting  Arcs. . .  .  590 
Ellipsis,  by  Intersecting  Lines...   588 

Ellipsis,  by  Ordinates 588 

Ellipsis,    Point    of    Contact   with 

Tangent,  To  Find 593 

Ellipsis,  Proportionate   Axes,    to 

Describe  with 594 

Ellipsis,  Trammel,  to    Find,  Axes 

Given 586 

Elliptical  Arch,  Joints,   Direction 

of 233 

English  Architecture,  Early n 

English  Cottage  Style  Extensive- 
ly Used 35 

j  England  and    France,  Fourteenth 
Century 12 


INDEX. 


665 


Entablature,  above  Columns  and 

Horizontal 14 

Entasis,  Swell  of  Shaft  of  Column    48 

Equal  Angles  Defined 349 

Equal  Angles,  Example  in 350 

Equal  Angles,  in  Circle 358 

Equal  Angles  (Geometry) 553 

Equilateral  Rectangle,  to  De- 
scribe   568 

Equilateral  Triangle  Defined  (Ge 

ometry) 545 

Equilateral  Triangle,  to  Construct 

(Geometry) 568 

Equilateral  Triangle,  to    Describe 

(Geometry) 566 

Eqilateral   Triangle,    to    Inscribe 

(Geometry) 569 

Equilateral  Triangle  (Polygons)..  445 

Eustyle  Defined 20 

Exponents,  Quantities  with  Nega- 
tive (Algebra) 423 

Extrados  of  an  Arch 52 

Face  Mould,  Accuracy  of,  Wind- 
ing Stairs 295 

Face  Mould,  Curves  Elliptical, 
Winding  Stairs 301 

Face  Mould,  Drawing  of,  Winding 
Stairs 296 

Face  Mould,  Sliding  of,  Winding 
Stairs 299 

Face  Mould,  Application  of,  Plat- 
form Stairs 275 

Face  Mould,  a  Simple,  Kell's 
Method  for 268 

Factors,  Multiplication  (Algebra)  409 

Factors,  Two,  Squaring  Difference 
of  (Algebra) 412 

Fibrous  Structure  of  Materials..  .      76 

Figure  Equal,  Given  Figure  (Ge- 
ometry)   575 

Figure,  Nearly  Elliptical,  To  Make 
(Geometry) 591 

Fillet  or  Amulet,  Classic  Mould- 
ing   323 

Fire-proof  Floors,  Action  of  Fire 
on JJ3 

Flanges,  Cast-iron  Girder 163 


Flanges,  Area  of,  Tubular  Iron 
Girder 155 

Flanges,  Area  of  Bottom,  Tubular 
Iron  Girder 159 

Flanges,  Load  at  Middle  of  Cast- 
iron  Girder,  Sizes 162 

Flanges,  Load  Uniform  on  Tubu- 
lar Iron  Girder,  Sizes 156 

Flanges,  Proportion  of,  Tubular 
Iron  Girder 157 

Flexure,  Compared  with  Rup- 
ture   84 

Flexureof  Rafter 205 

Flexure,  Resistance  to,  Defined. .   145 

Floor-arches,  How  Constructed. .   153 

Floor-arches,  Tie-rods,  Dwellings, 
Sizes 153 

Floor-arches,  Tie-rods,  First-class 
Stores,  Sizes 153 

Floor-beams,  Distance  from  Cen- 
tres, Sizes  Fixed 129 

Floor-beams,  Dwellings,  Safe 
Weight  for 126 

Floor-beams,  Dwellings,  Deflec- 
tion Given,  Sizes 127 

Floor-beams,  First-class  Stores, 
Deflection  Given,  Sizes 128 

Floor-beams,  Ordinary  Stores,  De- 
flection Given,  Sizes 127 

Floor-beams,  Stores,  Safe  Weight 
for 126 

Floor-beams,  Reference  to  Rules 
for  Sizes 125 

Floor-beams,  Reference  to  Trans- 
verse Strains 126 

Floor-beams,  Proportion  of 
Weight  on  All 130 

Floors  Constructed,  Single  or 
Double 124. 

Floors,  Fire-proof  Iron,  Action  of 
Fire  on 143 

Floors,  Framed,  Seldom  Used.  ..    124 

Floors,  Framed,  Openings  in 130 

Floors,  Headers,  Defined 130 

Floofs,  Ordinary,  Effect  of  Fire 
on 143 

Floors,  Solid  Timber,  Dwellings 
and  Assembly,  Depth 143 


666 


INDEX. 


Floors,  Solid   Timber,  First-class 

Stores,  Depth 144 

Floors,  Solid  Timber,  to  Make 

Fire-proof 143 

Floors,  Tail-beams  Defined 130 

Floors,  Trimmers  or  Carriage- 
beams  Defined 130 

Floors,  Wooden,  More  Fire-proof 

than  Iron,  Some* Cases 143 

Flyers  and  Winders,  Winding 

Stairs 251 

Foci  Defined 548 

Foci  of  Ellipsis  To  Find 586 

Foci  of  Ellipse,  Tangent 487 

Force  Diagram,  Load  on  Each 

Support 179 

Force  Diagram,  Truss,  Figs.  59, 

68  and  69 179 

Force  Diagram,  Truss,  Figs.  60,  70 

and  71 180 

Force  Diagram,  Truss,  Figs.  61, 

72  and  73 181 

Force  Diagram  Truss,  Figs.  63 

74  and  75 183 

Force  Diagram,  Truss,  Figs.  64, 

77  and  78 184 

Force    Diagram,  Truss,    Figs.  65, 

78  and  79 185 

Force   Diagram,  Truss,    Figs.  66, 

8oand8i 186 

Forces,  Parallelogram  of 59 

Forces,  Composition  of 66 

Forces,  Composition,   Reverse  of 

Resolution 67 

Forces,  Resolution  of 59 

Forces,    Resolution    of,    Oblique 

Pressure 59 

Foundations,  Description    of 47 

.  Foundations  in  Marshes,  Timbers 

Used 47 

Fractions,  Addition,  Like  Denom- 
inators     382 

Fractions  Added  and    Subtracted 

(Algebra) 403 

Fractions  Changed  by  Division!.   380 

Fractions  Defined 378 

Fractions,  Division,  Rule  for 389 

Fractions,  Division  by  Factors...  381 


Fractions  Divided  Graphically. . .   388 
Fractions  Graphically  Expressed.  378 
Fractions,  Improper,  Defined....  380 
Fractions,  Least  Common  Denom- 
inator  , 384 

Fractions,  Multiplication,  Rule. . .  387 

Fractions  Multiplied  Graphically.  386 

Fractions,  Numeratorand  Denom- 
inator    378 

Fractions,    Reduce  Mixed    Num- 
bers    381 

Fractions,   Reduction    to   Lowest 
Terms 384 

Fractions  Subtracted  (Algebra). . .  405 

Fractions,  Subtraction   Like    De- 
nominators   383 

Fractions,    Unlike    Denominators 
Equalized 383 

Framed    Beams,     Thickness     of, 
Rules 130 

Framed  Girder,  Bays  Defined 167 

Framed  Girders,  Compression  and 
Tension,  Dimensions 174 

Framed      Girders,     Construction 
and  Uses 166 

Framed     Girders,      Height     and 
Depth 167 

Framed  Girders,  Kinds   of  Pres- 
sure    173 

Framed  Girders,  Long,  Construc- 
tion of '.   174 

Framed  Girders,  Panels  on  Under 
Chord,  Table  of 167 

Framed  Girders,  Ties  and  Struts, 
Effect  of 174 

Framed  Girders,  Triangular  Pres- 
sure, Upper  Chord 168 

Framed  Girders,  Triangular  Pres- 
sure, Both  Chords 171 

Framed  Openings  in  Floors 130 

Framing  Beams,  Effect  of  Splic- 
ing   235 

Framing  Roof  Truss 237 

Framing  Roof  Truss,  Iron  Straps, 
Size  of 239 

France  and    England,  Fourteenth 
Century 12 

Friction,  Effect  of 82 


INDEX. 


667 


Frieze    between    Architrave    and 

Cornice 15 

Furring  Denned 125 

Gable,  a  Pediment  in  Gothic  Ar- 
chitecture      15 

Gaining  a  Beam  Denned 100 

General  Contents,  Table  of 613-624 

Geometrical    Progression    (Alge- 
bra)-  435 

Geometry,    Angles    of    Triangle, 

Three,  Equal  Right  Angle 354 

Geometry   Chords   Giving   Equal 

Rectangles 363 

Geometry  Denned 544 

Geometry,  Divide  a  Given  Line. .   555 
Geometry,  Divisions  in  Line  Pro- 
portionate    583 

Geometry,  Elementary 347 

Geometry,  Equal  Angles 553 

Geometry,     Equal     Angles,    Ex- 
ample   350 

Geometry,  Figure  Equal  to  Given 

Figure,  Construct 575 

Geometry,   Figure  Nearly  Ellipti- 
cal by  Compasses 591 

Geometry,  Measure  an  Angle. . . .  348 
Geometry  Necessary  in  Handrail- 
ing,  Stairs 257 

Geometry,  Opposite  Angles  Equal.  354 

Geometry,  Parallel  Lines 555 

Geometry,  a    Perpendicular,    To 

Erect 550 

Geometry.Perpendicular.let  Fall  a.  551 
Geometry,  Perpendicular,  Erect  at 

End  of  Line 551 

Geometry,  Perpendicular,  Let  Fall 

Near  End  of  Line 553 

Geometry,  Plane  Denned  (Stairs).  257 

Geometry,  Point  of  Contact 558 

Geometry,    Points,   Three   Given, 

Find  Fourth 559 

Geometry,  Right  Line  Equal  Cir- 
cumference      566 

Geometry,    Right    Lines,    Propor- 
tion Between 584 

Geometry,     Right     Lines,      Two 
Given,  Find  Third 582 


Geometry,     Square    Equal     Rec- 
tangle, To  Make 581 

Geometry,    Square    Equal    Given 

Squares,  To  make 577 

Geometry,  Square  Equal  Triangle, 

To  Make ..  582 

German  or  Romantic  Style,  Thir- 
teenth and  Fourteenth  Centuries,     n 
Girder,  Bearings,  Space  Allowed 

for 141 

Girder,     Bow-String,     Cast-Iron, 

Should  not  be  Used 163 

Girder,  Bow-String,  Substitute  for.  163 
Girder,    Construction   with   Long 

Bearings 140 

Girder,  Cast-Iron,  Load  Uniform, 

Flanges 163 

Girder,  Cast-Iron,  Load  at  Middle, 

Flanges 162 

Girder,  Cast-Iron,    Proper    Form 

of 161 

Girder  Denned,  Position  and  Use 

of 140 

Girder,  Different  Supports  for. . . .   140 

Girder,  Dwellings,  Sizes  for 141 

Girder,  Framed,  Bays  Denned 167 

Girders,     Framed,     Compression 

and  Tension,  Dimensions 174 

Girder,  Framed,  Construction  of..   140 
Girder,  Framed,  Construction  and 

Uses 166 

Girder,  Framed,   Construction  of 

Long 174 

Girder,    Framed,  Kinds   of  Pres- 
sure     173 

Girders,      Framed,     Height     and 

Depth 167 

Girders,  Framed,  Panels  on  Under 

Chord,  Table  of. 167 

Girders,  Framed,  Triangular  Pres- 
sure Upper  Chord 108 

Girders,  Framed,  Triangular  Pres- 
sure Both  Chords 171 

Girders,    Framed,    and     Tubular 

Iron 140 

Girders,  First-Class  Stores,  Sizes 

for 141 

Girders,  Sizes,  To  Obtain 141 


668 


INDEX. 


Girders,    Strengthening,    Manner 

of. 140 

Girders,  Supports,  Length  of,  Rule.  157 
Girders,  Tubular  Iron,  Construc- 
tion of 154 

Girders,    Tubular   Iron,   Area    of 

Flange.  Load  at  Middle 154 

Girders,   Tubular   Iron,    Area    of 

Flange,  Load  at  any  Point 155 

Girders,    Tubular    Iron,    Area  of 

Flange,  Load  Uniform 156 

Girders,  Tubular  Iron,  Dwellings, 

Area  of  Bottom  Flange 159 

Girders,  Tubular  Iron,  First-Class 

Stores,  Area  of  Bottom  Flange..   160 
Girders,  Tubular  Iron,  Rivets,  Al- 
lowance for 157 

Girders,   Tubular  Iron,    Flanges, 

Proportion  of 157 

Girders,  Tubular  Iron,    Shearing 

Strain 15? 

Girders,  Tubular  Iron.Web,  Thick- 
ness of i58 

Girders,  Weakening,  Manner  of. .  140 
Girders,  Wooden,  Objectionable.  154 
Girders,  Wooden,  Supporting, 

Manner  of 154 

Glossary  of  Terms 627-637 

Gothic  Arches 51 

Gothic  Buildings,  Roofs  of 55 

Gothic  and  Norman  Roofs,  Con- 
struction of 178 

Gothic  Pillar,  Form  of 48 

Gothic  Style,  Characteristics  of. . .     12 

Goths,  Ruins  Caused  by 12 

Granular  Structure  of  Materials. .     76 

Gravity,  Centre  of,  Position 71 

Gravity,  Centre  of,  Examples,  and 

Rule  for 71 

Grecian  Architecture,  History  of.       6 

Grecian  Art,  Elegance  of 27 

Grecian  Moulding,  Cyma-Recta..  327 
Grecian  Moulding,  Cyma-Re- 

versa. . .  -. 328 

Grecian   Moulding,   Echinus  and 

Cavetto 327 

Grecian  Moulding,  Scotia 326 

Grecian  Moulding,  Torus 326 


Grecian    Orders   Modified  by  the 

Romans 27 

Grecian  Origin  of  the  Doric  Or 

der 16 

Grecian  Origin  of  Ionic  Order  ...  16 

Grecian  Style  in  America 13 

Grecian  Styles,  their  Different 

Orders 16 

Greek  Architecture,  Doric  Order 

Used 19 

Greek  Building 35 

Greek  Moulding,  Form  of 325 

Greek,  Persian,  and  Caryatides 

Orders 24 

Greek  Style  Originally  in  Wood..  14 
Greek  Styles  Only  Known  by 

Them 16 

Groined  or  Complex  Vault 52 

Halle  du  Bled,  Paris,  Dome  of. . .     54 

Halls  of  Justice,  N.  Y.  C.,  Speci- 
men of  Egyptian  Architecture. .  8 

Handrailing,  Circular  Stairs 278 

Handrailing,  Platform  Stairs.    ...  269 

Handrailing,  Platform  Stairs,  Face 
Mould 264 

Handrailing,  Platform  Stairs, 
Large  Cylinder 271 

Handrailing  Stairs,  Geometry 
Necessary 257 

Handrailing  Stairs,"  Out  of  Wind" 
Defined. 257 

Handrailing  Stairs,  Tools  Used. .   257 

Handrailing,  Winding  Stairs. .256,  289 

Handrailing  Winding  Stairs,  Bal- 
usters Under  Scroll 310 

Handrailing,  Winding  Stairs, 
Centres  in  Square 308 

Handrailing,  Winding  Stairs,  Face 
for  Scroll 311 

Handrailing,  Winding  Stairs,  Fall- 
ing Mould 310 

Handrailing,  Winding  Stairs,  Gen- 
eral Considerations 258 

Handrailing,  Winding  Stairs, 
Scroll  for 308 

Handrailing,  Winding  Stairs, 
Scroll  at  Newel 309 


INDEX. 


669 


Handrailing,      Winding       Stairs, 

Scroll  Over  Curtail  Step 309 

Handrailing,      Winding       Stairs, 

Scroll  for  Curtail  Step 310 

Headers,  Breadth  of. 130 

Headers  Defined 130 

Headers,  Mortises,  Allowance  for 

Weakening  by 131 

Headers,   Stores   and    Dwellings, 

Same  for  Both 132 

Hecadecagon,    Complete    Square 

(Polygons) 458 

Hecadecagon,    Radius   of  Circles 

(Polygons) 455 

Hecadecagon,  Rules  (Polygons). .   459 
Hecadecagon,     Side      and     Area 

(Polygons) 457 

Height  and   Projection,  Numbers 

of  an  Order 16 

Hemlock,  Weight  per  Foot  Super- 
ficial    177 

Heptagon  Defined 546 

Herring-bone  Bridging  Defined...   137 

Hexagon  Defined 546 

Hexagon,  To  Inscribe 569 

Hexagons,  Radius  of  Circles 447 

Hexastyle,  Intercolumniation. . . .     20 
Hindoo     Architecture,    Ancient, 

Character  of 30 

Hip-Rafter,  Backing  of 216 

Hip-Roofs,    Diagram   and  Expla- 
nation   215 

History  of  Architecture 44 

Hogged  Ridge  in  Roof  Truss 238 

Homologous     Triangles    (Geom- 
etry)   362 

Homologous      Triangles      (Ratio 

and  Proportion) 370 

Hooke's  Theory  of  an  Arch 50 

Hooke's    Theory,    Bridge    Arch, 

Pressure  on.  ...   51 

Hooke's  Theory,  Catenary  Arch. .     51 
Horizontal  and  Inclined  Roofing, 

Weight 190 

Horizontal  Pressure  on  Roof,  To 

Remove 74 

Horizontal  Thrust  in  Beams 72  i 

Horizontal  Thrust,  Tendency  of. .     88  i 


Hut,  Original  Habitation 13 

Hydraulic  Method,  Testing 
Woods 80 

Hyperbola  Defined 548,  585 

Hyperbola,  Height,  To  Find,  Base 
and  Axis  Given 585 

Hyperbola  by  Intersecting  Lines.   595 

Hypothenuse,  Equality  of  Squares 
(Algebra)  416 

Hypothenuse,  Formula  for  (Trig- 
onometry)   516 

Hypothenuse,  Side,  To  Find  (Ge- 
ometry)   579 

Hypothenuse,  Right  Angled  Tri- 
angle (Geometry) 355 

Hypothenuse,  Triangle  (Trigo- 
nometry)   518 

Ichnographic  Projection,  Ground 

Plan 37 

Improper  Fractions  Defined 380 

India  Ink  in  Drawing 540 

Inertia,  Moment  of,  Defined 145 

Inking-in  Drawing . .   542 

Inside  Shutters  for  Windows,  Re- 
quirements   319 

Instruments  in  Drawing 540 

Intercolumniation  Defined 17 

Intercolumniation  of  Orders 20 

Intrados  of  Arch 52 

Ionic  Order,  Character  of. 16 

Ionic  Order,  Grecian  Origin  of. . .      16 
Ionic  Order  Modified  by  the  Ro- 
mans. ...i 27 

Ionic  Order,  Origin  of 20 

Ionic   Order,   Suitable   for  What 

Buildings 20 

Ionic  Volute,  To  Describe  an. ...     20 
Iron  Beams,  Breaking  Weight  at 

Middle 148 

Irpn  Beams,  Deflection,  To  Find, 

Weight  at  Middle 147 

Iron  Beams,  Deflection,  To  Find, 

Weight  Uniform 150 

Iron  Beams,  Dimensions,  To  Find, 

Weight  any  Point 149 

Iron  Beams,  Dimensions,  To  Find, 
Weight  Uniform 149 


6  ;o 


INDEX. 


Iron  Beams,  Dwellings,  Distance 
from  Centres 151 

Iron  Beams,  First-Class  Stores, 
Distance  from  Centres 152 

Iron  Beams,  Rectangular  Cross- 
Section 145 

Iron  Beams,   Rolled,  Sizes 145 

Iron  Beams,  Safe  Weight,  Load 
any  Point 148 

Iron  Beams,  Safe  Weight,  Load 
Uniform 151 

Iron  Beams,  Table  IV 146 

Iron  Beams,  Weight  at  Middle, 
Deflection  Given 146 

Iron  Fire-Proof  Floors,  Action  of 
Fire  On 143 

Iron  Straps,  Framing,  to  Prevent 
Rusting 239 

Irregular  Polygon,  Trigon  (Geom- 
etry)   546 

Isosceles  Triangle  Denned 545,  584 

Italian  Architecture,  Thirteenth, 
Fourteenth,  and  Fifteenth  Cen- 
turies   12 

Italian  Use  of  Roman  Styles 13 

Italy,  Tuscan  Order  the  Principal 
Style 30 

Jack-Rafters,  Location  of 212 

jack-Rafters  and  Purlins  in  Roof.  211 
Jack-Rafters,  Weight  per  Superfi- 
cial Foot 189 

Joists  and  Studs  Defined 174 

Jupiter,  Temple  of,  at  Thebes,  Ex- 
tent of 33 

Kell's      Method,     Simple      Face 

Mould,  Stairs 268 

Keystone  for  Arch,  Position  of. ...     50 
King-Post,   Bad    Framing,   Effect 

of ...  237 

King-Post,  Location  of 213 

King-Post  in  Roof 54 

Lamina  in  Girders  Defined 174 

Lancet  Arch 51 

Lateral  Thrust  in  Arch 52 


Laws  of  Pressure 57 

Laws   of   Pressure,  Inclined,  Ex- 
amples       57 

Laws  of  Pressure,  Vertical,  Exam- 
ples      57 

Leaning    Tower     or     Campanile, 

Twelfth  Century 12 

Length,    Breadth,    or    Thickness, 

Relation  to  Pressure 78 

Lever,     Breadth     or     Depth,    To 

Find in 

Lever,    Deflection   as   Relating    to 

Beam 1 19 

Lever,  Deflection,  Load  at  End..  120 
Lever,  Deflection,  Load  Uniform.  121 
Lever,  Deflection,  Breadth  or 

Depth,  Load  at  End 121 

Lever,     Deflection,     Breadth     or 

Depth,  Load  Uniform 122 

Lever,  Deflection,  Load  Required.  122 
Lever  Formula,  Proportionals  in 

(Algebra) 421 

Lever   Load    Uniformly     Distrib- 
uted    in 

Lever,  Load  at  One  End no 

Lever     Principle      Demonstrated 

(Ratio) 375 

Lever,  Support,  Relative  Strength 

of  One no 

Light-Houses,  Eddystone  and  Bell 

Rock 48 

Line  Defined  (Geometry) 544 

Lines,  Divisions  in,  Proportionate 

(Geometry) 583 

Lintel,  Position  of 49 

Lintel,  Strength  of 49 

Load,  per  foot,  Horizontal 192 

Load  on  Roof  Truss,  per  Superfi- 
cial  Foot 189 

Load  on  Tie-Beam,  Ceiling,  etc. .   190 

Lock  Rail  for  Doors,  Width 316 

Logarithms  Explained  (Algebra)..   425 

Logarithms,  Examples 426 

Logarithms,    Sine    and    Tangents 

(Polygons) 464 

Lombard,  Byzantine  Style 10 

Lombard  Style,  Seventh  Century.  10 
London  Bridge,  Piles,  Age  of 229 


INDEX. 


67I 


Materials,  Cohesive  Strength  of. .     76 
Materials,    Compression,    Resist- 
ance to. 77 

Materials,  Cross-strain,  Resistance 

to 77 

Materials,  Structure  of 76 

Materials,  Tension,  Resistance  to.     77 
Materials     Tested,    General     De- 
scription       80 

Materials,  Weights,  Table  of. .    . .  654 
Major  and  Minor  Axes  of  Ellipse 

Denned 481 

Marshes,  Foundation  for  Timbers 

in 47 

Mathematics  Essential 347 

Maxwell,  Prof.  I.  Clerk,  Diagrams 

of  Forces,  etc 165 

Memphis,    Pyramids  of,  Estimate 

of  Stone  in 33 

Minster,  Tower  of  Strassburg n 

Minutes,    Sixty  Equal    Parts,    to 

Proportion  an  Order 15 

Mixed  Numbers  in  Fractions,  To 

Reduce 381 

Modern   Architecture,    First   Ap- 
pearance of 9 

Modern  Tuscan,  Appropriate  for 

Buildings 30 

Moment  of  Inertia  Defined 145 

Mono-triglyph,  Explanation  of  the 

Word 19 

Monuments,  Ancient,  Their  Archi- 
tects        6 

Moorish  and  Arabian  Styles,  An- 
tiquities of ii 

Mortises,  Proper  Location  of —  .   100 
Mortising,      Beam,      Effect       on 

Strength  of 100 

Mortising  Beam  at  Top,  Injurious 

Effect  of 100 

Mortising  Beam,  Effect  of 231 

Mortising,  Beam,  Position  of 236 

Mortising  Headers,  Allowance  for 

Weakening  131 

Moulding,    Classic,     Astragal  or 

Bead 323 

Moulding,    Classic,    Annulet    or 
Fillet 323 


Mouldings,  Classic  Architecture.  323 
Moulding,  Classic,  Cavctto  or 

Cove 323 

Moulding,  Classic,  Cyma-Recta. .  324 
Moulding,  Classic,  Cyma-Reversa.  324 

Moulding,  Classic,  Ogee 324 

Moulding,  Classic,  Ovolo 323 

Moulding,  Classic,  Scotia 323 

Moulding,  Classic,  Torus 323 

Mouldings,  Common  to  all  Or- 
ders   324 

Mouldings  Defined 323 

Mouldings,  Diagrams  of 330 

Mouldings, Doors,  Rule  for  Width.  317 
Moulding,  Grecian,  Cyma-Recta.  327 
Moulding,  Grecian,  Cyma-Rc- 

versa 328 

Moulding,   Grecian    Echinus   and 

Cavetto 327 

Mouldings,  Greek,  Form  of. 325 

Mouldings,    Grecian    Torus    and 

Scotia 326 

Mouldings,  Modern 331 

Moulding,  Modern,  Antae  Cap...   334 
Mouldings,  Mbdern  Interior,  Dia- 
grams   332 

Mouldings,  Modern,  Plain 333 

Mouldings,Names,  Derivations  of.  324 

Mouldings,  Profile  Defined 326 

Mouldings,  Roman,  Forms  of. ..  .  325 
Mouldings,  Roman, Comments  on.  329 
Mouldings,  Roman,  Ovolo  and 

Cavetto 329 

Mouldings,Uses  and  Positions  of.  324 

Multiplication  (Algebra) 408 

Multiplication,    Plus    and    Minus 

(Algebra) 415 

Multiplication,  Three  Factors  (Al- 
gebra)  408 

Multiplication,  Fractions 387 

Newel  Cap,  Form  of,  Winding 
Stairs 312 

Nicholson's  Method,  Plane 
Through  Cylinder  (Stairs) 259 

Nicholson's  Method,  Twists  in 
Stairs 259 

Nonagon  Defined 546 


672 


INDEX". 


Normal  and  Subnormal  in  Para- 
bola   496 

Norman  and  Gothic  Construction 
of  Roofs 178 

Norman  Style,  Peculiarities  of. . .      n 

Nosing  and  Tread,  Position  in 
Stairs 241 

Oblique  Angle  Defined 544 

Oblique  Pressure,  Resolution  of 

Forces . . . . 59 

Oblique  Triangle,  Difference  Two 

Angles  (Trigonometry) 523 

Oblique  Triangle,  First  Class 

(Trigonometry) , 520 

Oblique  Triangles,  First  Class, 

Formulae  (Trigonometry) 531 

Oblique  Triangles,  Second  Class 

(Trigonometry) 522 

Oblique  Triangles,  Second  Class, 

Formulae  (Trigonometry) 532 

Oblique  Triangles,  Third  Class 

(Trigonometry) 526 

Oblique  Triangles,  Third  Class, 

Formula;  (Trigonometry) 534 

Oblique  Triangles,  Fourth  Class 

(Trigonometry) 528 

Oblique  Triangles,  Fourth  Class, 

Formulae  (Trigonometry) 534 

Oblique  Triangles,  Two  Sides 

(Trigonometry) 521 

Oblique  Triangles,  Sines  and 

Sides  (Trigonometry) 519 

Obtuse  Angle  Denned 349,  544 

Obtuse  Angled  Triangle  Denned.  545 
Octagon,  Buttressed,  Find  Side 

(Geometry) 571 

Octagon  Defined  546 

Octagon,  Diagonal  of  Square 

Forming 357 

Octagon,  Inscribe  a  (Geometry). .  570 

Octagon,  Rules  (Polygons) 451 

Octagon,  Radius  of  Circles  (Poly- 
gons)   449 

Octastyle,  Intercolumniation 20 

Ogee  Mouldings,  Classic 324 

Opposite  Angles  Equal  (Geome- 
try)   354 


Order     of    Architecture,      Three 

Principal  Parts 14 

Orders  of  Architecture,    Persians 

and  Caryatides 24 

Ordinates  to  an  Arc  (Geometry).  .  563 

Ordinates,  Circle,  Rule  for 471 

Ordinates  of  Ellipse 491 

Ostrogoths,  Style  of  the 9 

Oval,  To  Describe  a  (Geometry). .  591 

Ovolo,  Classic  Moulding 323 

Ovolo,  Roman  Moulding 329 

Paper,  The,  in  Drawing,  Secure  to 

Board , .  .  537 

Pantheon  at  Rome,  Dome  of,  and 

Walls 53 

Pantheon  and   Roman   Buildings, 

Walls  of 49 

Parabola,  Arcs  Described  from...  503 

Parabola,  Area,  Rule  for 509 

Parabola,  Axis  and   Base,  to  find 

(Geometry) 585 

Parabola,  Curve,  Equations  to...  .  493 

Parabola  Defined 492 

Parabola  Defined  (Geometry).. 548,  585 

Parabola,  Diameters 497 

Parabola   Described    from    Ordi- 
nates    504 

Parabola  Described  from  Diame- 
ters   507 

Parabola  Described  from  Points..  502 

Parabola  of  Dome  Computed 219 

Parabola,  General  Rules 499 

Parabola  by  Intersecting  Lines. . .  594 
Parabola  Mechanically  Described.  500 
Parabola,    Normal    and    Subnor- 
mal    496 

Parabola,  Ordinate  Defined 496 

Parabola,  Subtangent 496 

Parabola,  Tangent 493 

Parabola,    Vertical    Tangent    De- 
fined   495 

Parabolic     Arch,      Direction      of 

Joints   234 

Parallel  Lines  Defined 544 

Parallel  Lines  (Geometry) 555 

Parallelogram,  Construct  a 576 

Parallelogram  Defined 545 


INDEX. 


673 


Parallelogram  Equal  to  Triangles, 

To  Make.. 576 

Parallelogram   of  Forces,    Strains 

by 165 

Parallelograms    Proportioned    to 

Bases  (Geometry) 360 

Parallelogram      in       Quadrangle 

(Geometry) 364 

Parallelogram,  Same  Base  (Geom- 
etry)    352 

Parameter  Defined 548 

Parameter,  Axes  (Ellipse) 485 

Parthenon  at  Athens,  Columns  of.     48 
Partitions,  Bracing  and  Trussing.   176 

Partitions,  How  Constructed 174 

Partition,   Door  in  Middle,  Con- 
struction     175 

Partition,  Doors  at  End, Construc- 
tion of 176 

Partition,    Great    Strength,    Con- 
struction    176 

Partitions,  Location  and  Connec- 
tion  •.   175 

Partitions,  Materials,  Quality  of. .   175 
Partitions,  Plastered,  Proper  Sup- 
ports for. 175 

Partitions,  Pressure  on.  Rules 177 

Partitions,  Principal,  of  what  Com- 
posed    175 

Partitions,  Trussing  in,  Effects  of.   175 
Pedestal,    a    Separate     Substruc- 
ture      14 

Pediment,     Triangular     End     of 

Building 15 

Pencil  and  Rulers,  Drawing 540 

Pentagon  Defined 546 

Pentagon,  Circumscribed  Circles 

(Polygons) 463 

Perpendicular  Height  of  Roof,  To 

find 579 

Perpendicular,  Erect  a 550 

Perpendicular,  Erect  a,  at  End  of 

Line 551 

Perpendicular,  Let  Fall  a 551 

Perpendicular,  Let  Fall  a,  at  End 

of  Line 553 

Perpendicular      Style,     Fifteenth 
Century 12 


Perpendicular  in  Triangle  (Poly- 
gons)    440 

Persians,  Origin  and  Description 

of 24 

Persians  and  Caryatides,    Orders 

Used  by  Greeks 24 

Piers,  Arrangement,  in  City  Front 

of  House 44 

Piers,   Bridges,  Construction  and 

Sizes 228 

Piles,  London  Bridge,  Age  of. ...  229 
Pine,    White,    Beams,    Table     of 

Weights  for. 177 

Pisa,  Cathedral  of,  Eleventh  Cen- 
tury        12 

Pisa,    Cathedral    of,    Erection    in 

1016 12 

Pise  Wall  of  France 49 

Pitch  Board,  To  Make,  for  Stairs.   247 

Pitch  Board,  Winding  Stairs 252 

Plane  Defined 257 

Plane  Defined  (Geometry) 544 

Plank.Weightof.on  Roof,  per  foot.  189 
Plastering,    Defective,    To     what 

Due 174 

Plastering,  Strength  of 174 

Plastering,  Weight  per  foot 177 

Platform    Stairs,    Baluster,    Posi- 
tion of 250 

Platform  Stairs  Beneficial 240 

Platform  Stairs,  Cylinder  of. 248 

Platform  Stairs,  Cylinder,  Lower 

Edge 249 

Platform  Stairs,  Face  Mould,  Ap- 
plication of  Plank 273 

Platform     Stairs,     Face     Mould, 

Handrailing  in 264 

Platform  Stairs,  Face  Mould,  Sim- 
ple Method 267 

Platform      Stairs,     Face     Mould, 

Moulded  Rails 274 

Platform  Stairs,  Face  Mould,  Ap- 
plication of 275 

Platform  Stairs,  Face  Mould  With- 
out Canting  Plank 272 

Platform  Stairs,  Handrail  to 269 

Platform  Stairs,  Handrailing  Large 
Cylinder 271 


674 


INDEX. 


PAGE  I 

Platform    Stairs,    Railing    Where 

Rake  Meets  Level 272 

Platform    Stairs,  Twist-Rail,   Cut- 
ting of 277 

Platform  Stairs,  Wreath  of  Round 

Rail 267 

Point  of  Contact  (Geometry) 558 

Point  Denned  (Geometry) 544 

Pointed  Style,  Ecclesiastical  Arch- 
itecture      ii 

Polygons,  Angles  of 462 

Polygons,  Circumscribed  and  In- 
scribed Circles,  Radius  of 460 

Polygons  Defined  (Geometry). . .  .   546 
Polygons,  Equilateral  Triangle. ..  445 

Polygons,  General  Rules 461 

Polygons,  Irregular,  Trigon  (Ge- 
ometry)    546 

Polygons,  Perpendicular  in  Tri- 
angle    440 

Polygon,  Regular,  Defined  (Geom- 
etry)  546 

Polygons,  Regular,  To   Describe 

(Geometry) 573 

Polygons,  Regular,  To  Inscribe  in 

Circle  (Geometry) 572 

Polygons,    Sum   and    Difference, 

Two  Lines 439 

Polygons,  Table  Explained 466 

Polygons,  Table  of  Multipliers. ..  465 
Polygons,  Triangle,  Altitude  of..  442 
Polygonal  Dome,  Shape  of  Angle- 
Rib 223 

Posts,  Area,  To  Find 86 

Posts,  Diameter,  To  Find 92 

Posts,  Rectangular,  Safe  Weight..     92 
Posts,  Rectangular.To  Find  Thick- 
ness       94 

Posts,  Rectangular,  Breadth  Less 

than  Thickness 96 

Posts,     Rectangular,     To     Find 

Breadth 95 

Posts,  To  Find  Side 93 

Posts,  Slender,  Safe  Weight  for. .     91 
Posts,  Stout. Crushing  Strength  of.     89 

Pressures  Classified 85 

Pressure,  Oblique,  Resolution  of 
Forces 59 


Pressure,      Triangular,     Framed 

Girders 171 

Pressure,  Upper  Chord,  Triangu- 
lar Girder 168 

Prisms  Cut  by  Oblique  Plane 259 

Prisms  and  Cylinders,  Stair-Build- 
ing  257 

Prisms  Defined  (Stairs) 257,  259 

Prism,  Top,  Form  of,  in  Perspec- 
tive    259 

Profile  of  Mouldings  Defined. . . .   326 
Progression,   Arithmetical  (Alge- 
bra)  432 

Progression.Geometrical  (Algebra)  435 
Projection  and  Height,  Members 

of  Orders  of  Architecture 16 

Protractor,  Useful  in  Drawing.  ..   541 
Purlins  and  Jack-Rafters  in  Roof.  211 

Purlins,  Location  of 212 

Pyramids  of  Memphis,  Amount  of 

Stone  in 33 

Pycnostyle,  Explanation  of 20 

Quadrangle  Defined 545 

Quadrangle  Equal  Triangle 353 

Quadrant  Defined 547 

Quantities,    Addition    and     Sub- 
traction (Algebra) 424 

Quantities,  Division  of  (Algebra).  424 
Quantities,  Multiplication  of  (Al- 
gebra)  424 

Queen-Post,  Location  of 213 

Queen-Post  in  Roof 54 

Radials  of  Rib  in  Bridge 226 

Radials  of  Rib  for  Wedges 226 

Radicals,  Extraction  of  (Algebra).  425 

Radius  of  Arc,  To  Find 561 

Radius  of  Circle  Defined 547 

Rafters,  Braces,  etc.,  Length,  To 

Find 580 

Rafters,  Least  Thrust,  Rule  for. . .  62 

Rafters,  Length  of,  To  Find 578 

Rafters,  Liability  to  Crush  Other 

Materials 205 

Rafters,  Liability  to  Being  Crushed  205 

Rafters,  Liability  to  Flexure 205 

Rafters,  Minimum  Thrust  of 62 


INDEX. 


6/5 


PAGE 

Rafters  in  Roof,  Effect  of  Weight 
on 179 

Rafters  in  Roof,  Strains  Subjected 
to , 205 

Rafters  and  Tie-Beams,  Safe 
Weight 87 

Rafters,  Uses  in  Roof 54 

Rake  in  Cornice  Matched  with 
Level  Mouldings 344 

Railing,  Platform  Stairs  Rake 
Meets  Level 272 

Ratio  or  Proportion,  Equals  Mul- 
tiplied    367 

Ratio  or  Proportion,  Equality  of 
Products 370 

Ratio  or  Proportion,  Equality  of 
Ratios 367 

Ratio  or  Proportion  Equation, 
Form  of 367 

Ratio  or  Proportion,  Examples..  .   366 

Ratio  or  Proportion,  Four  Propor- 
tionals, to  Find 377 

Ratio  or  Proportion,  Homologous 
Triangles 370 

Ratio  or  Proportion,  Lever  Prin- 
ciple in 372 

Ratio  or  Proportion,  Lever  Prin- 
ciple Demonstrated 375 

Ratio  or  Proportion,  Multiply  an 
Equation 368 

Ratio  or  Proportion,  Multiply  and 
Divide  One  Number 368 

Ratio  or  Proportion,  Rule  of 
Three 366 

Ratio  or  Proportion,  Steelyard  as 
Example  in 371 

Ratio  or  Proportion,  Terms  of 
Quantities 367 

Ratio  or  Proportion,  Transfer  a 
Factor 369 

Rectangle  Defined 545 

Rectangle,  Equilateral,  To  De- 
scribe   5°8 

Rectangular  Cross-Section,  Iron 
Beams 145 

Reduction  Cubic  Feet  to  Gallons, 
Rule 653 

Reduction  Decimals,  Examples..   647 


Reflected  Light,  Opposite  of 
Shade 611 

Regular  Polygon  in  Circle,  To  In- 
scribe (Geometry) 572 

Regular  Polygon  Defined  (Geom- 
etry)   546 

Regular  Polygons,  To  Describe 
(Geometry) 573 

Resistance,  Capability  of 86 

Resistance  to  Compression,  Ap- 
plication of  Pressure 85 

Resistance  to  Compression, 
Crushing  and  Bending 85 

Resistance  to  Compression,  Mate- 
rials   77 

Resistance  to  Compression,  Pres- 
sure Classified 85 

Resistance  to  Compression  in 
Proportion  to  Depth 101 

Resistance  to  Compression,  Stout 
Posts,  Rule 89 

Resistance  to  Compression,  Table 
of  Woods 79 

Resistance  to  Cross-Strains 77 

Resistance  to  Cross-Strains  De- 
fined   99 

Resistance  to  Deflection,  Rule... .    113 

Resistance  Depending  on  Com- 
pactness and  Cohesion 78 

Resistance  Depending  on  Loca- 
tion, Soil,  etc 79 

Resistance  to  Flexure  Defined. .  .    145 

Resistance  Inversely  in  Propor- 
tion to  Length 102 

Resistance  to  Oblique  Force 206 

Resistance,  Power  of,  Hew  Ob- 
tained   78 

Resistance,  Proportion  to  Area. . .     86 

Resistance,  Strains,  To  What  Due.     78 

Resistance  to  Tension  Greatest 
in  Direction  of  Length 81 

Resistance  to  Tension,  Proportion 
in  Materials 81 

Resistance  to  Tension,  Table  of 
Materials 82 

Resistance  to  Tension,  Materials.      77 

Resistance  to  Tension,  Results 
from  Transverse  Strains  . .  82 


6;6 


INDEX. 


PAGE 

Resistance  to  Transverse  Strains,  Roman  Architecture,  Ruins  of. .  .     n 

Table  of. 83    Roman    Architecture,    Excess    of 

Resistance  to  Transverse  Strains,          I      Emichment 46 

Description  of  Table 84    Roman  Building 26 

Resistance  Variable  in  One  Ma-  Roman  Composite  and  Corinthian 

terial.... 79        Orders.. .     28 

Reticulated  Walls .  < 49  j  Roman  Decoration 27 

Rhomboid  Defined 546    Roman  Empire,  Overthrow  of. ...     13 

Romans,  Ionic  Order  Modified  by     27 

Roman  Moulding,  Cavetto 329 

Roman  Mouldings,  Comments  on.  329 
Roman  Moulding,  Ovolo 329 


Rhombus  Defined ' . . .  .   545 

Ribbed  Bridge,  Area  of  Rule 225 

Ribbed  Bridge,  Built 224 

Ribbed  Bridge,  Least  Rise,  Table 

of 224 

Right  Angle  Defined 348,  544 

Right  Angle    in    Semicircle  (Ge- 
ometry)     355 

Right  Angle,  To  Trisect  a 554 

Right  Angled  Triangle  Defined  . .   545 
Right   Angled   Triangle,    Squares 

on  (Algebra) 417 

Right    Angled    Triangles  (Trigo- 
nometry)    510 

Right  Angled  Triangles,  Formula 

for  (Trigonometry) 530 

Right  Lines  (Geometry) 584 

Right  Line  Equal  Circumference.   566 
Right  Lines,   Mean  Proportionals 

Between 584 

Right    Lines,   Two    Given,    Find 

Third 582 

Right  Lines,   Three  Given,  Find 

Fourth 583 

Right  or  Straight  Line  Defined. . .    544 

Right  Prism  Defined  (Stairs) 257 

Risers,  Number  of,  Rule  to  Ob- 
tain (Stairs) 246 

Rise  and  Tread  (Stairs) 241 

Rise    and  Tread,   Connection   of 

(Stairs) 248 

Rise  and  Tread,  Blondel's  Method 

of  Finding  (Stairs) 242 

Rise    and    Tread,    Table    of,   for 

Shops  and  Dwellings  (Stairs). . .   245 
Rise  and  Tread,  To  Obtain  (Wind- 
ing Stairs) 251 

Rolled  Iron  Beams,  Extensive  Use 


Roman  Mouldings,  Forms  of.. ...  325 


49 
26 

13 


of. 


161 


Roman  Architecture  Defined. ....       7 


Roman  Pantheon,  etc.,  Walls  of. . 
Roman  Styles  of  Architecture. . . . 
Roman  Styles  Spread  by  the  Ital- 
ians  

Romantic  or  German  Style,  Thir- 
teenth and  Fourteenth  Centu- 
ries    ii 

Rome,  Ancient  Buildings  of. 12 

Rome    and   Greece,    Architecture 

of 8 

Roof,  The 54 

Roofs,      Ancient      Norman     and 

Gothic,  Construction 178 

Roof  Beams,  Weight  per  Super- 
ficial Foot 189 

Roof,  Brace  in,  Rule  Same  as  for 

Rafter 208 

Roofs,  Construction  of. 55 

Roof  Covering,  Mode  of 188 

Roof  Covering,  Weights,  Table  of.  191 

Roof,  Curb  or  Mansard 54 

Roofs,  Diagrams  and  Description 

of 212 

Roof,  Gothic  Buildings 55 

Roofs,  Gothic  and  Norman  Puild- 

ings,   Construction 178 

Roofs,  Hip,  Diagram  and  Exj 

nation 215 

Roof,  Hip 54 

Roof,  Horizontal  Pressure,  To  Re- 
move from 74 

Roof,  Jack-Rafters  and  Purlins.. .   211 
Roof,  King-Post  in 54 


Roof,   Load  per  Foot  Horizontal, 


Rule, 


IQ2 


INDEX. 


677 


Roof,  Load,  Total  per  Foot  Hori- 
zontal, Rule 197 

Roofs,   Modern,  Trussing  Neces- 
sary    178 

Roofs,  Norman  and  Gothic  Build- 
ings,    178 

Roof,  Pent,  To  Find 54 

Roof,    Perpendicular    Height,  To 

Find 579 

Roof  Plank,    Weight    per   Super- 
ficial Foot 189 

Roof,  Planning  a 188 

Roof,  Pressure  on 55 

Roof,  Queen-Post  in 54 

Roof,  Rafters  in 54 

Roof,  Sagging,  To  Prevent.  ......     54 

Roof,  Slope   Should  Vary  Accord- 
ing to  Climate 191 

Roof  Supports.  Distance  between.   189 
Roof,     Suspension      Rods,     Safe 

Weight  for 210 

Roof,  Tie-Beam  in 54 

Roof,    Tie-Beam,   Tensile   Strain, 

Rule ,.   204 

Roof  Timbers,  Mortising 55 

Roof  Timbers,  Scarfing  of 55 

Roof  Timbers,  Splicing  of 55 

Roof  Timbers,  Strains  by  Parallel- 
ogram of  forces 198 

Roof  Timbers,  Strain  Shown  Ge- 
ometrically  199,  202 

Roof  Truss,  Arched  Ceiling 214 

Roof  Truss,    Elevated    Tie-Beam 

Objectionable 214 

Roof  Truss,  Elevating  Tie-Beam, 

Effect  of 187 

Roof  Truss,  Force  Diagram,  Figs. 

59,  68,  and  69 179 

Roof  Truss,  Force  Diagram,  Figs. 

60,  70,  and  7 r 180 

Roof  Truss,  Force  Diagram,  Figs. 

61,  72,  and  73 181 

Roof  Truss,  Force  Diagram,  Figs. 

63,74,  and  75  183 

Roof  Truss,  Force  Diagram,  Figs. 

64,  77,  and  78 184 

Roof  Truss,  Force  Diagram,  Figs. 

66,  80,  and  81 186 


Roof  Truss,  Load  on 189 

Roof  Trusses,  Strains,  Effect  of,  on 

Different 179 

Roof  Truss,  Weights,  Table  of,  per 

Superficial  Foot 189 

Roof  Truss,  Weight  per   Superfi- 
cial Foot 190 

Roof,  Trussing  in 54 

Roof  Trussing,  Designs  for 178 

Roof  Trussing,  Framing  for 237 

Roof  Trussing,  Hogged  Ridge....   238 
Roof  Trussing,   King-Post,  Effect 

of  Bad  Framing  on 237 

Roofs,  United  States 55 

Roof,  Vertical   Pressure  of  Wind 

on,  Effect  of. 194 

Roof,  Snow,  Weight  per  Horizon- 
tal Foot 193 

Roof  Weight  on  Rafter,  Effect  of..   179 
Roof,  Wind,  Horizontal  and  Verti- 
cal Pressure  of 193 

Roofing,  Weight  of  Horizontal  and 

Inclined 190 

Roofing,   Weight    per    Superficial 

Foot 190 

Roots,  Cubes,  and  Squares,  Table 

of 638-645 

Round  Post,  Area  of 90 

Rubble  Walls 48 

Rulers  and  Pencil  in  Drawing....  540 
Rupture  Compared  with  Flexure.  84 
Rupture,  Crushing,  Safe  Weight..  89 
Rupture,  Sliding,  Safe  Weight. ...  87 
Rupture,  Transverse,  Safe  Weight.  86 
Rusting  Iron  Framing  Straps,  To 
Prevent 239 

Safe  Load  for  Material 81 

Safe  Weight,  Allowance  for 84 

Safe  Weight  at  Any  Point,  Rule. .  106 

Safe  Weight,  Beam  at  Middle 103 

Safe  Weight,  Bending 91 

Safe  Weight,  Beam,  Breadth  of, 

To  Find 104 

Safe  Weight,  Beam,  Depth,  To 

Find 104 

Safe  Weight,  Breadth  or  Depth,  To 

Find,  Load  at  Middle 106 


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INDEX. 


PAGE 

247 
240 

247 


Stairs,  Space  for  Timber  and  Plas- 
ter  

Stairs,  Stone,  Public  Building 

Stairs,  String  of,  To  Make 

Stairs,  Tread,  To  Find,  Rise  Given 
242,  246 

Stairs,  Tread  and   Riser  Connec- 
tion     248 

Stairs,  Width,  Rule  for 241 

Stairs,     Winding,     Balusters     in 
Round  Rail 313 

Stairs,  Winding,  Bevels  in  Splayed 
Work ...  314 

Stairs,     Winding,    Blocking    Out 
Rail.. 301 

Stairs,    Winding,    Butt-joint     on 
Handrail 303 

Stairs,  Winding,    Butt-joint,  Cor- 
rect Lines  for 307 

Stairs,   Winding,     Diagrams    Ex- 
plained     ..   263 

Stairs,  Winding,  Face  Mould,  Ac- 
curacy of 295 

Stairs,    Winding,     Face     Mould, 
Application 297 

Stairs,    Winding,     Face    Mould, 
Care  in  Drawing 295 

Stairs,    Winding,     Face     Mould, 
Curves  Elliptical 301 

Stairs,    Winding,      Face     Mould 
for 290,  293 

Stairs,     Winding,    Face     Mould, 
Round  Rail 303 

Stairs,  Winding,  Face  Mould  for 
Twist \ . .  291 

Stairs,       Winding,       Flyers     and 
Winders 251 

Stairs,    Winding,      Front    String, 
Grade  of 253 

Stairs,  Winding,  Handrailing. 256,  289 

Stairs,  Winding,  Handrailing,  Bal- 
usters Under  Scroll 310 

Stairs,      Winding,      Handrailing, 
Centres  for  Square 368 

Stairs,     Winding,       Handrailing, 
Face  Mould  for  Scroll 311 

Stairs,  Winding,  Handrailing,  Fall- 
ing Mould  for  Raking  Scroll. . .  310 


Stairs,  Winding,  Handrailing,  Gen- 
eral Considerations 258 

Stairs,  Winding,  Handrailing, 
Scrolls  for 308 

Stairs,  Winding,  Handrailing, 
Scroll  Over  Curtail  Step 309 

Stairs,  Winding,,  Handrailing, 
Scroll  for  Curtail  Step 310 

Stairs,  Winding,  Scroll  at  Newel.  309 

Stairs,  Winding,  Illustrations  by 
Planes 261 

Stairs,  Winding,  Moulds  for 
Quarter  Circle 255 

Stairs,  Winding,  Newel  Cap,  Form 
of 312 

Stairs,  Winding,   Objectionable.  .    240 

Stairs,  Winding,  Pitch  Board,  To 
Obtain 252 

Stairs,  Winding,  Rise  and  Tread, 
To  Obtain 251 

Stairs,  Winding,  Sliding  of  Face 
Mould. 299 

Stairs,  Winding,  String,  To  Ob- 
tain    252 

Stairs,  Winding,  Timbers,  Posi- 
tion of 252 

Stairs  and  Windows,  How.  Ar- 
ranged    42 

Stiles  of  Windows,  Allowance  for.   319 

St.  Mark,  Tenth  or  Eleventh  Cen- 
tury   12 

Stone  Bridge  Building,  Truss 
Work 232 

Stone  Bridge,  Building   Arch....   230 

Stone  Bridge,  Centres  for,  Con- 
struction   229 

Stone  Bridge,  Pressure  on  Arch 
Stones 230 

Stop  for  Doors 317 

Stores,  Floor  Beams,  Safe  Weight.  126 

Stores,  Ordinary,  Floor-Beams, 
Sizes,  To  Find 127 

Stores,  First-Class,  Floor-Beams, 
Sizes,  To  Find 128 

St.  Paul's,  London,  Dome  of 54 

St.  Peter's,  Rome,  Fourteenth  and 
Fifteenth  Centuries 12 

Straight  or  Right  Line  Defined. . .   544 


INDEX. 


681 


Strains,  Cross,  Resistance  to 77 

Strains  on  Domes,  Tendency  of. .   219 

Strains  Exceed  Weights 61 

Strains,  Graphic  Representation..  165 
Strain  Greatest  at  Middle  of  Beam.  105 
Strains  by  Parallelogram  of 

Forces 165 

Strains,  Practical    Method  cf  De- 
termining      62 

Strains  of  Rafter  in  Roof 205 

Strains,  Resistance,  To  What  Due.     78  | 
Strain    on    Roof  Timbers    Shown 

Geometrically 190 

Strains  on  Roof  Timbers  Geomet- 
rically Applied 202  j 

Strains  on  Roof  Timbers,  Parallel- 
ogram of  Forces 198  ; 

Strain,    Shearing,    Tubular     Iron 

Girder 157 

Strain  Unequal,  Cause  of 83 

Straps,  Iron,  Roof  Truss   239  j 

Strassburg,  Cathedral  of 12  j 

Strassburg,    Towers   of  the  Min- 
ster      II  I 

Strength    and    Stiffness   of   Mate- 
rials..-  ."...     78  j 

Structure  of  Materials 76  | 

Struts  Denned 173  j 

Struts  and  Ties 68  i 

Struts   and    Ties,    Difference    Be- 
tween       69  : 

St.  Sophia,  Sixth  Century 12  i 

Stucco  Cornice  for  Interior 340  ; 

Studs  and  Joists  Defined 174  ' 

Styles,   Grecian,  Only  Known   by 

Them 16  ! 

Stylobate,    Substructure  'for   Col- 
umns      14  • 

Subnormal  and  Normal   (in  Para- 
bola)    496  I 

Subtangent,  Parabola 496 

Subtangent  of  Ellipse  Defined...  486 
Subtraction   and   Addition  (Alge- 
bra)   398 

Superficies  Defined  (Geometry)...  544 
Supports,  Girders,  Length,  Rule..  157 

Supports,  Position  of 65 

Supports,  Inclination  of,  Unequal.     60 


Suspension  Rods,  Location  in 
Roof 212 

Suspension  Rods  in  Roof,  Safe 
Weight 210 

Symbols  Chosen  at  Pleasure  (Al- 
gebra)   395 

Symbols,  Transferring  (Algebra).   399 

Systyle,  Explanation  of 20 

Table  of  Circles 649-652 

Table  of  Contents 6^3-624 

Table  of  Capacity  of  Wells,  Cis- 
terns, etc 653 

Table    of     Squares,    Cubes,    and 

Roots 638-645 

Table  of  Woods,  Description  of. .     80 

Tail-Beams  Defined 130 

Tanged  Curve,  To  Describe  (Ge- 
ometry)   565 

Tangent  to  Axes,  Ellipse . . . .  485 

Tangent  Defined 547 

Tangent  with  Foci,  Ellipse 487 

Tangent  to  Ellipse.  To  Draw 592 

Tangent  at  Given  Point  in  Cir- 
cle  557 

Tangent  at  Given   Point,  Without 

Centre 557 

Tangent  of  Parabola 493 

Tangents  and    Sines,    Logarithms 

(Polygons) 464 

Temples  Built  in  the  Doric  Style.      19 

Temple,  Doric,  Origin  of  the 17 

Temple  of  Jupiter  at  Thebes 33 

Tenons  and    Splices,    Knowledge 

Important   88 

Tensile  Strain,  Area  of  Piece,  To 

Find 99 

Tensile  Strain,  Compressed  Ma- 
terial   ioo 

Tensile  Strain,  Condition  of  Sus- 
pended Piece 98 

Tensile  Strain,  Safe  Weight 96 

Tensile  Strain,   Safe   Weight,  To 

Compute 97 

Tensile  Strain,  Sectional  Area,  To 

Obtain 97 

Tensile  Strain,  Suspended,  M»te- 
terial  Extended ioo 


682 


INDEX. 


PAGE 

Tensile  Strain  on  Tie-Beam  in  Roof 

Truss 204 

Tensile  Strain,  Weight  of  Suspend- 
ed Piece 98 

Tensile  Strength  of  Cast  Iron 161 

Tension  and  Compression,  Framed 

Girders 174 

Tension,  Resistance  to 77 

Tension,  Resistance   to,  Table  of 

Materials 82 

Tension,    Resistance   to,    Results 

Obtained 82 

Tension,  Resistance  to,  Proportion 

in  Materials 81 

Tent,  Habitation  of  the  Shepherd.  13 
Testing  Machine,  Description  in 

Transverse  Strains 80 

Testing       Materials,       Hydraulic 

Method 80 

Testing  Materials,  Dates  of 80 

Testing  Materials,  Manner  of. ...     80 

Tetragon  Defined  546 

Tetragon,       Radius     of     Circles 

(Polygons) 446 

Tetrastyle,  Intercolumniation 20 

Thebes,  Thickness  of  Walls  at. . .     33 

Thrust,  Horizontal 63 

Thrust,  Horizontal,  Examples. ...  64 
Thrust,  Horizontal,  Tendency  of..  88 
Tie-Beam  in  Ceiling,  Load  on...  190 
Tie-Beam  and  Rafter,  Safe  Weight.  87 

Tie-Beam  in  Roof 54 

Tie-Beam  in  Roof,  Tensile  Strain.  204 

Tie-Rods,  Diameter,  To  Find 164 

Tie-Rods,  Floor   Arches,    Dwell- 
ings    153 

Tie-Reds,    Floor   Arches,    First- 

Class  Stores 153 

Tie-Rods,  Wrought  Iron 164 

Ties  Defined 173 

Ties  and  Struts,  To  Distinguish..     69 
Ties  and  Struts,  Framed  Girders..   174 
Ties  and  Struts,  Principles  of. ...     68 
Ties,  Timbers  in  a  State  of  Ten- 
sion      68 

Titus,  Composite  Arch  of 28 

Trimme/,  Breadth,  To  Find,  Two 
Sets  Tail-Beams 134 


Top  Rail,  Doors,  Width,  Rule 317 

Torus,  Classic  Moulding 323 

Torus,  Grecian  Moulding. 326 

Tower  of  Babel,  History  of 5 

Towers  of  the  Minster,  Strassburg.     n 

Transverse  Axis  Defined 548 

Transverse    Strains,    Compressed 

and  Extended,  Material 100 

Transverse  Strains,  Defined 99 

Transverse    Strains,    Explanation 

of  Table  III 101 

Transverse        Strains,        Greater 

Strength  of  One  Piece 101 

Transverse  Strains,  Neutral   Line 

Defined 100 

Transverse  Strains,  Proportion  to 

Breadth 101 

Transverse      Strains,      Hatfield's, 

Reference  to.  .80,  121,  133,  138, 

143,  144,  145,  146,  148 
Transverse  Strains,  Resistance  to, 

Table  of 83 

Transverse  Strains,  Description  of 

Table 84 

Transverse   Strains,    Strength  Di- 
minished by  Division 101 

Trapezoid   Defined 546 

Trapezium  Defined 546 

Tread,      To    Find,    Rise     Given 

(Stairs) 242,246 

Tread    and     Nosing,   Position   of 

(Stairs) 241 

Tread  and  Rise,  To  Find,  Winding 

Stairs 251 

Tread  and  Rise,  To  Find,  Blon- 

del's  Method 242 

Tread  and  Rfcse,  Table  for  Shops  \ 

and  Dwellings 245 

Tread    and   Riser,  Connection    of 

(Stairs) 248 

Triangle,  Altitude  of  (Polygons).   442 
Triangles,  Base,  Formula  for  (Trig- 
onometry)    516 

Triangle,  Construct  a  (Geometry).  587 
Triangle,    Construct    Equal-Sided 

(Geometry) 575 

Triangle  Defined 545 

Triangle,  Examples  (Geometry).. .  350 


INDEX. 


683 


PAGE 

Triangles,  Equal  Altitude 361 

Triangle  Equal  Quadrangle 353 

Triangles,  Equation  of  (Trigo- 
nometry)   515 

Triangles,  Homologous  (Geom- 
etry)   362 

Triangles,  Hypothenuse,  Formula 
for 516 

Triangles,  Hypothenuse,  To  Find 
(Trigonometry) 518 

Triangles,  Perpendicular,  To  Find 
(Trigonometry) 517 

Triangle  or  Set-Square  in  Draw- 
ing    539 

Triangle  or  Set-Square,  Use  of. . .   541 

Triangles,  Terms  Denned  (Trigo- 
nometry)   512 

Triangles,  Three  Angles  Equal 
Right  Angle 354 

Triangles,  Value  of  Sides  (Trigo- 
nometry)   516 

Trigon,  Irregular  Polygons  (Ge- 
ometry)   546 

Trigon,  Radius  of  Circle  (Poly- 
gons)  443 

Trigon,  Rule  (Polygons) 441 

Trigonometry,  Oblique  Triangles, 
Two  Angles 523 

Trigonometry,  Oblique  Triangles, 
Two  Sides 521 

Trigonometry,  Oblique  Triangles, 
First  Class 520 

Trigonometry,  Oblique  Triangles, 
Second  Class 522 

Trigonometry,  Oblique  Triangles, 
Third  Class 526 

Trigonometry,  Oblique  Triangles, 
Fourth  Class 

Trigonometry,  Oblique  Triangles, 
Sines  and  Sides 

Trigonometry,  Oblique  Triangles, 
Formula,  First  Class 531 

Trigonometry,  Oblique  Triangles, 
Formula,  Second  Class 532 

Trigonometry,  Oblique  Triangles, 
Formula,  Third  Class 534 

Trigonometry,  Oblique  Triangles, 
Formula,  Fourth  Class 534 


S*\ 

519 


Trigonometry,  Right  Angled  Tri- 
angles    510 

Trigonometry,   Right  Angled  Tri- 
angles, Third  Side,  To  Find...    511 

Trigonometry,  Right  Angled   Tri- 
angle, Formula 530 

Trigonometry,  Tables 513 

Trigonometry,     Triangles,     Base, 
Formula  for 516 

Trigonometry,    Triangles,     Equa- 
tions of 515 

Trigonometry,  Triangles,  Hypoth- 
enuse, Formula 516 

Trigonometry,  Triangles,  Hypoth- 
enuse, To  Find 518 

Trigonometry,  Triangles,  Perpen- 
dicular, To  Find 517 

Trigonometry,   Triangles,    Terms 
Denned 512 

Trigonometry,  Triangles,  Value  of 
Sides 516 

Trimmer     or     Carriage      Beam, 
Breadth,  To  Find 132 

Trimmer  or  Carriage  Beams  De- 
nned     130 

Trimmer,  One    Header,   Breadth, 
To  Find,  Dwellings  and  Stores.   133 

Trimmer,  Well-Hole    in    Middle, 
Breadth,  To  Find 136 

Trisect  a  Right  Angle 5^4 

Truss,  Diagram  of. 200 

Truss,  Force  Diagrams,  Figs.  59, 

68  and  69 179 

Figs.  60,  70  and  71 iSo 

Figs.  61,  72  and  73 181 

Figs.  63,  74  and  75 183 

Figs.  64,  77  and  78 184 

Figs.  65,  78  and  79 185 

Figs.  66,  80  and  81 186 

Truss,  Roof,  Framing  for 237 

Truss,  Roof,  Iron  Straps 239 

Truss,    Weight,    per     Horizontal 
Foot,  To  Find 192 

Truss  Work,  Stone  Bridge  Build- 
ing   232 

Trussing   and    Framing,    Gravity 
and  Resistance 76 

Trussing  Partitions,  Effect  of 175 


684 


INDEX. 


Trussing  Roofs,  Effect  of 178 

T-Square,  How  to  Make 539 

Tubular  Iron  Girder,  Area  of  Bot- 
tom Flange,  Dwellings 159 

Tubular  Iron  Girder,  Area  of  Bot- 
tom Flange,  First-Class  Stores.  160 

Tubular  Iron  Girder,  Arc  of 
Flange,  Load  at  Middle 154 

Tubular  Iron  Girder,  Area  of 
Flange,  Load  Any  Point 155 

Tubular  Iron  Girder,  Area  of 
Flange,  Load  Uniform 156 

Tubular  Iron  Girder,  Flanges, 
Proportion  of. 157 

Tubular  Iron  Girder,  Construction 

of 154 

Tubular  Iron  Girder,  Rivets,  Al- 
lowance for 157 

Tubular    Iron     Girder,     Shearing 

Strain 157 

Tubular  Iron  Girders,  Web  of. . .    158 
Tuscan,  Modern,  Appropriate  for 

Buildings 30 

Tuscan  Order,  Introduction  of  the.  30 
Tuscan  Order,  Principal    Style  in 

Italy 30 

Twelfth  Century,  Buildings  in  the.     n 

Twist  Rail,  Platform  Stairs 277 

Twists,  Stairs,  Nicholson's  Method 
for 259 


Undecagon  Defined 

United  States,  Roofs  in. 


Vault,  Simple,  Groined  or  Com- 
plex  

Ventilation,  Proper  Arrangement 
for...!. 

Versed  Sine  of  Arc,  To  Find 

Vertical  Pressure  of  Wind  on 
Roof. 

Vertical  Tangent  of  Parabola  De- 
fined  

Volutes,  To  Describe  the 

Voussoir  of  an  Arch. . , 


Wall,  The... 
Walls,  Coffer. 


546 

55 


52 

45 
56i 

194 

495  ! 

20  * 

52  I 

| 

48  I 
49! 


PAGE 

Walls,    Construction  and  Forma- 
tion        48 

Walls,  Eddystone  and  Bell  Rock 

Lighthouses 48 

Walls,  Egyptian,  Massiveness  of..     33 

Walls,  Modern  Brick 49 

Walls   of    Pantheon    and    Roman 

Buildings 49 

Walls  of  Pantheon  at  Rome 53 

Walls,  Pise,  of  France 49 

Walls,  Reticulated 49 

Walls,  Rubble -. ...     48 

Walls,  Strength  of. 48 

Walls,  Various  Kinds 49 

Walls,  Wooden 49 

Weakening  Girder,  Manner  of. . .    140 
Web   of    Tubular     Iron     Girder, 

Thickness  of 158 

Weight  of  Materials  for  Building 

Table  of 654-656 

Wells,    Cisteins,    etc.,    Table    of 

Capacity 653 

White    Pine,  Weights   of    Beams 

Table  of 177 

Wind,  Greatest  Pressure,  per  Su- 
perficial Foot 90 

Wind  on  Roof,  Effect  of  Vertical 

Pressure 194 

Wind    on    Roof,   Horizontal    and 

Vertical  Pressure 193 

Winders  in  Stairs,    How  to  Place 

the 42 

Winders  and  Flyers,  Stairs 251 

Windows,  Arrangement  of 44 

Windows,  Circular  Headed 320 

Windows,     Circular   Headed,  To 

Form  Soffit 321 

Windows,  Dimensions,  To  Find.   318 

Window-Frame,  Size  of 318 

Windows,  Front  of  Building,  Ef- 
fect of 320 

Windows,      Heights,      Table    of, 

Width  Given 320 

Windows,  Height  from  Floor. . .  .  320 
Windows,    Inside    Shutters,    Re- 
quirement   319 

Windows,     Position     and    Light 
from.. 317 


INDEX. 


685 


Windows  and  Stairs,  How  Ar- 
ranged    42 

Windows,  Stiles,  Allowance  lor..   319 

Windows,  Width  Uniform,  Height 
Varying  . . . : 319 

Winding  Stairs,  Balusters  in 
Round  Rail 313 

Winding  Stairs,  Bevels  in  Splayed 
Work 314 

Winding  Stairs,  Blocking  Out 
Rail.. 301 

Winding  Stairs,  Butt  Joint,  Posi- 
tion of. 303 

Winding  Stairs,  Butt  Joint 307 

Winding  Stairs,  |Diagram  of,  Ex- 
plained    263 

Winding  Stairs,  Face  Mould  for 
290,293 

Winding  Stairs,  Face  Mould,  Ac- 
curacy of. 295 

Winding  Stairs,  Face  Mould,  Ap- 
plication of 297 

Winding  Stairs,  Face  Mould, 
Curves  Elliptical 301 

Winding  Stairs,  Face  Mould, 
Drawing 296 

Winding  Stairs,  Face  Mould, 
Round  Rail 303 

Winding  Stairs,  Face  Mould,  Slid- 
ing of. 299 

Winding  Stairs,  Face  Mould  for 
Twist 291 

Winding  Stairs,  Flyers  and  Wind- 
ers   251 

Winding  Stairs,  Front  String, 
Grade  of. 253 

Winding  Stairs,  Handrailing 
. ; 256,  289 


Winding  Stairs,  Handrailing,  nal- 

usters  Under  Scroll 310 

Winding  Stairs,  Handrailing,  Cen- 
tres in  Square 308 

Winding  Stairs,  Handrailing,  Face 

Mould  for  Scroll. .., 311 

Winding  Stairs,  Handrailing,  Fall- 
ing Mould. . . 310 

Winding      Stairs,       Handrailing, 

General  Considerations 258 

Winding       Stairs,       Handrailing, 

Scrolls  for 308 

Winding       Stairs,      Handrailing, 

Scroll  Over  Curtail  Step 309 

Winding      Stairs,       Handrailing, 

Scroll  for  Curtail  Step 310 

Winding      Stairs,        Handrailing, 

Scrolls  at  Newel 309 

Winding    Stairs,    Illustrations    by 

Planes 261 

Winding  Stairs,  Moulds  for  Quar- 
ter Circle 255 

Winding  Stairs,  Newel  Cap,  Form 

of 312 

Winding  Stairs  Objectionable. . . .  240 
Winding   Stairs,   Pitch  Board,  To 

Obtain 252 

Winding  Stairs,  Rise  and    Tread, 

To  Obtain 251 

Winding  Stairs,  String,  To  Obtain.  252 
Winding    Stairs,    Timbers,    Posi- 
tion of. 252 

Wood,  Destruction  by  Fire 37 

Wooden  Beams,  Use  Limited 154 

Woods,      Hydraulic    Method      of 

Testing 80 

Wreath  for  Round  Rail,  Platform 
Stairs 267 


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